How can I protect my hands when using power tools?
The wearing of gloves can be a terrible idea when using many power tools. In fact, it can be quite dangerous for the reasons you describe, although kevlar gloves might be a good idea when you are around a chainsaw. They might even be a good idea if you are working with sharp chisels. Even so, I would suggest that it makes far more sense to begin using good, safe practices.

If your fingers are getting too close to a moving powered object, then you are doing it WRONG! If you are taking risks with your fingers, then learn safe working practices!

NEVER WEAR LOOSE FITTING CLOTHING OF ANY SORT WHEN WORKING WITH POWER TOOLS. Gloves are part of this.

Never get close enough to a blade for there to be a problem. Use a push stick or block on table saws. Use hold downs of all sorts. Use a table saw with a saw stop.

If you are using a drill, why are your fingers touching the drill bit?

If you can't control your fingers and other body parts, then perhaps chess is a better activity for you? There are always people willing to be paid to do your home repair projects for you. Or, learn to do those projects safely.

When I go into my wood shop, I wear a short sleeve shirt. No gloves. I am aware of the tools that I use. I use proper push blocks and hold downs where appropriate. And when my fingers get near a moving object, I ALWAYS know exactly where they are and I am in complete control of the situation, or I find a different way to do the job.

Finally, if you are tired, then stop working. There is always another day to finish the job, but you have only one life, only so many fingers, eyes, etc.