I think the main thing is 5th edition - it's really good. I briefly tried 4th edition and it just didn't feel right - way too combat heavy and video-gamey.
Then 5th edition came out and it was immediately obvious that it had that 1st edition feel but with all the bugs worked out. I play it with my kids and it's awesome.
Would you care to share some 'bugs'? I'm always pretty interested in those things, despite never having had a.. session(?) myself. I know of one where you used a Spiked Chain, made yourself a giant(?) so you got a big boost in combat stats, and because of how the chain works you'd almost always succeed in tripping someone, and when they got up got a free counter attack that again could trip them. Another one I do not know the method of (except that it has to do with Simulacra and Wish) which let's you basically make infinite copies of yourself.
In our Pathfinder campaign, we have a level 10 evocation (fireball, pew pew) halfing (I think?) wizard with a raven familiar. Said wizard has permanent Reduce person (so to be a tiny creature) and their familiar has permanent Enlarge Person to make it a small creature. The wizard sits in a backpack on the crow, and for roleplaying purposes (wizard lost a bet to familiar) the raven does all the talking and casting.
Before feats, the raven can fly 40' a round. Between that and items to allow for empowered and maximized fireball and scorching ray (I think?), our wizard regularly drops between 60 and 120 damage per round.
I suspect GP is referring more to complexities in the gameplay, not loopholes (though I'm sure 5e has fewer of those, by nature of being less rules-heavy and more reliant on the GM's interpretation). For example, weird corner cases like grappling rules have been greatly simplified. So have skills, weapon selection, and combat rules in general.
As late as 3.5e and Pathfinder, you could easily spend 5 minutes figuring out how to resolve a single 6-second action according to the rules. Say I'm grappling someone (complex rules) and move them across rough terrain (5-foot step or not?) while threatened by an enemy combatant (attacks of opportunity) with while under a Bard buff (pluses to defense)… figuring out who attacks whom when and how can easily involve flipping to many different sections of the rule book. Whereas in 5e half those rules don't exist and the remainder are simplified.
My understanding (not from experience) is that previous editions had even more complex rules than 3.5e, and 4e simplified the rules in the "wrong direction" (away from role-playing).
In first edition the Armor Class went from 10 downwards eg AC -2 was awesome. And you had to look up tables (before 2nd edition added the slightly simpler THAC0). That was changed such that numbers always go up and similarly, higher dice rolls are always better (in some circumstances in 1e you wanted lower numbers).
The concept of advantage or disadvantage. Sometimes you'd have to do a bunch of calculations eg -4 for being invisible, +2 for this, -1 for that, etc. Now, you just figure out whether someone has an advantage or disadvantage. If neither, it's a normal dice roll. If advantage, roll 2 dice and take the highest number. If disadvantage, roll 2 dice and take the lower.
Some classes were crazy powerful, and others weren’t. It was easy to get into a situation where one player would dominate the game. And not really on purpose. Now the classes are much more even. Everyone gets a chance to shine.
Honestly, forget about rules. Just tell a story and let them guide it. You can occasionally throw in some d20 rolls if you want to introduce some risk of failure, and from there perhaps start build in some bonuses which let them customise their characters. But the main thing is just collaborative storytelling (and bad jokes and accents) and the best role-playing sessions I've ever had have involved no dice rolls at all.
I used to play like this with my friends in high school. We'd just start talking about our characters whenever we were together. Occasionally throw in some dice. It was a lot of fun and could be done anywhere and anytime. Create a simple fantasy setting, dream up some characters, and explore your world.
Graham Walmsley's short book "Play Unsafe" is a good introduction to running a game and incorporating advice from improv to improve your group's storytelling.
I would strongly suggest you take a look at "The Lazy DM" by Sky Flourish [0]. He has other books that are pretty good also. This method of running games will get you going immediately and takes almost no preparation. You will stumble a couple of times when you first try it, because it feels unnatural versus drawing everything out ahead of time. But once you get used to it, you'll never go back, and you'll be able to run any number of games with almost no preparation. That isn't to say you shouldn't prepare some things ahead of time: Great adventures and fantastic dungeons are always best created when you have the spark. But this method of running games will make you much, much more efficient with your (precious) time. I have a personal library of drop-in dungeons and adventures that I've created which I wrap with this method of running games. And if I didn't have this method, I don't think I would be able to run, since I just don't have the time as an adult with a regular job. Good luck!
This looks really interesting — the only reason I don't run more games more often is the time burden. At a glance it appears to be focused on the various D&D editions + Pathfinder, and was published pre-5th (though I can't find a publication date on the site).
Is it generalised/generalisable enough for use with other source material and RPGs? In other words, is it D&D specific?
Yes, absolutely generalized. It basically outlines a generic "world/story engine" using index cards and a kind of "rule of threes". I've used this in all kinds of systems: Shadowrun, D&D, Pathfinder, AD&D, Star Wars d20, etc. It's totally usable in any system. It cuts down on the preparation time required by orders of magnitude, seriously.
One tip that I use in every game that is incredibly useful and isn't in this book (or Sly Flourish's other great book, "Dungeon Master Tips") is to have three NPCs who represent "Money, Power, and Fame" and are always on the move. Their storylines develop even if the players are not interacting with them. Remember "Fame, Money, Power" and create a single NPC to represent each of these three things. For example, you could have a local baron who is hungry for power, an evil guildmaster who is trying to expand his influence over local trade routes for money, and a retired, good-hearted adventurer who helps out new adventurers at the local adventurer's guildhall because he enjoys the fame and recognition.
Each of these three NPCs will have some immediately obvious conflicts which will generate storylines you can very easily improvise. I don't even need to give you an example since you can probably think of some conflict in those three examples just off the top of your head. These three NPCs also have easy access to each of those three things (money, power and fame respectively) and typically, at least some of your players will be motivated by this to some degree. That makes for an easy carrot-and-stick for players. Do your players need money for an expensive spell component to save the mayor's daughter? You might find the guildmaster willing to pay a handsome reward for completing a small side quest, for example having the players bluff or sneak their way into the baron's palace during a party and steal some important documents. The local baron of the land can probably give you the power to raise a small militia to stop a rampaging band of orcs in the area. Does the king's court have no idea who you are? It turns out the retired adventurer might be willing to write a note on your behalf, if only you could do him a small favor...
And the important thing is that you should always be moving these three NPCs along their life paths even if the players aren't around to see it. This is very easy to track with your index cards. And if one of these NPCs is killed or retires or somehow drops out of the game, you should come up with another to replace them. Perhaps the evil guildmaster has the baron assassinated, and now that he's had a taste of power after his guild is given control of the barony, he becomes your "Power" NPC, and a rival guildmaster becomes the new "Money" NPC. See how easy that was? And you can do this in your sleep really, so it works out great for the method described in The Lazy DM.
The starter set is pretty good to get you up and going with DMing.
The best bit of advice I can give is to let the kids be creative and don't be too fussed about the rules. It's like improv - any idea is a gift - even if it's an outlandish idea, let them do it, make up some difficulty score and when the dice come up with a failure, try to be as creative and funny as possible in describing the outcome.
If you're not married to the idea of D&D you can check other systems that are more focused on a "pick up and play" style such as "Fate Accelerated" from Evil Hat.
depending on the age of the kids, you might be interested in facilitating No Thank You, Evil. It's designed to be run by an older kid, but parents can GM as well. I've not played, but I have heard good reviews.
From what I remember, the main criticism of the d20 systems was that it was too flexible, and that the game ended up very build-oriented. Some of the design decisions behind 4th edition were an attempt to streamline it with lessons from online RPGs, but part of the reasoning behind it was that this streamlining would move the focus away from the character sheets a bit - by not writing everything down in rules, things would be more up to DM interpretations.
I cannot say to which degree the game systems actually helped in practice; I switched groups between editions, and my 4th edition group was 90% improv theatre people. That there would be more roleplaying was pretty inevitable.
I also haven't played 5th edition so I don't know the changes there.
Then 5th edition came out and it was immediately obvious that it had that 1st edition feel but with all the bugs worked out. I play it with my kids and it's awesome.