I'm not really an expert, but one obvious thing is you need to take non-pure metals and purify them. Recyclable batteries fail when the chemicals instead chemically combine with something other what you intend them to, crystalize, or otherwise become not the pure metal you start with. That impure metal is not conceptually different from ore, and the same process to turn ore into battery material must be done.
Note that ore is more complex as you need to remove a lot more non-battery stuff, while in a used battery what you want is still in fairly high concentration, just not in the form a battery needs.
Generally yeah - it is both willing to give up and take back electrons pretty easily (so, discharging and charging) but in the modern era another really important factor is that it balances having these properties and also being very light relative to the amount of energy it can hold in a battery. Good for portable applications like cell phones, cars, etc.
Lithium is at the top, so you get the most voltage per molecule.
Then there’s other factors like discharge recharge, temperatures, all that, but lithium is basically the best if you can get the other factors to play nice too:
That was covered in my college chemistry class, but I took it 25 years ago and don't really remember the details. Electro negativity comes to mind, but I might have the terms wrong. In any case the laws of chemistry apply.
Note that ore is more complex as you need to remove a lot more non-battery stuff, while in a used battery what you want is still in fairly high concentration, just not in the form a battery needs.