Can I wear non-fingerless gloves regularly without it looking weird?

Go to a therapist and get to the root of your problem. You need to stop chewing your fingernails. Then after that let them grow out and go get them manicured.

No you can not have gloves on all the time without people noticing and wonder why. It will look very odd.

There are products to use to help you from biting your nails. 

not at all. use it as you want

No. People will think you have a skin disease or something. Stop biting your nails.

Ladies used to wear gloves all the time but that'sl ong our of fashion.  it would look odd.

It looks completely normal to wear disposable gloves in public now, regardless of what season it is.  

Sure, especially now because people will just think you're trying to avoid Covid.