How do I best protect my hardwood floors from damage by moving chair legs
There are Teflon glides that can be screwed on. You might try those on one chair to see how they work out for you. Because they screw on, then can be easily unscrewed. I would drill a pilot hole - though your existing nail holes might suffice.



Teflon is a proprietary name for PTFE, so search also for screw-on PTFE furniture glides.

Teflon is slippery, so a disadvantage may be that your chairs might scoot away from you if you are not careful when sitting down. There's no getting away from the fact that staying-put requires friction which means abrasion.