My understanding is that retail trading volume is several orders of magnitude too low to be responsible for swings in most stocks (such as airline stocks), and certainly not the market as a whole. Random low volume penny or bankrupt stocks are a whole different story and retail speculation certainly drives the price.
You’d be surprised how little volume it takes to make prices move intraday. Market makers are not rigidly stupid and sell you as much as you want at a fixed price. If you continue to buy aggressively, they will raise the prices while you are buying.
Also retail isn’t really well defined in terms of dollar amounts. there are some extremely wealthy people classified as retail. And they don’t need to be as ultra conservative with their trading as institutional traders because they are trading their own money.
In small stocks, when using leverage, they can influence things. I can’t find it now (on mobile) but I saw an analysis of HTZ that showed almost all of its rally was due to retail.
Call buying forces the options market maker to delta hedge by purchasing the underlying.
Example: As of market close on Friday, a June 19, 2020 expiration HTZ $3 strike call option cost .55 ($55) and has a delta of .56; if I were to buy a 10 lot for $550, the options market maker would be short 560 deltas and would purchase 560 shares of HTZ to hedge their short call position. This can push the share price up quite a bit if there’s non-stop call buying.
The converse is also true for put options, buying puts from an options market maker forces them to sell the underlying to delta hedge.
I worked at firm that let you trade on leverage a while back. They ended up owning so much of a small stock they couldn’t unwind the position after the leveraged client sold. I think they had to slowly unwind it and take on the risk.