Is the drain cover in my shower sufficient?
Hair going down the drain is inevitable and no solution will completely prevent it.

Some drains can be replaced from above while others cannot. Unless you are having a chronic problem with clogging due to hair, I think you are making more work for yourself than necessary.  If you try to remove it and can't, or don't install the new one correctly you'll go from letting a bit of hair down the drain to a full-scale leak.

It's also not going to stop draining completely out of the blue; it will gradually start draining slower, so it's very unlikely that this will become an emergency one night.  I'd just wait and see. If you have a problem, then you'll obviously need to unclog it, otherwise, enjoy your shower and don't worry too much!