Yes, that absolutely can happen. However, it requires improper hygiene (not washing hands before touching face). If a person is not going to wash their hands before touching their face, how likely is it that they will properly utilize gloves? In the context of preventing spread/infection, proper glove usage requires changing gloves (or sanitizing) every time you touch a new surface - otherwise you're potentially spreading the virus every time you touch something. There are some places where this could be reasonably practiced (i.e gas stations), but in pretty much any other public environment, the waste generated would be enormous for a very small gain in protection.
To put things in perspective, in order to contract COVID from a surface, you first need to have an infected individual spread fluids on the surface. The fluids that they spread must contain more than enough viral load overcome your immune system. Sufficient viral load must then survive on the surface long enough for you to come into contact with it. Then, sufficient viral load must enter your body somehow (mouth, nose, eyes). The chances of that happening are extremely low - especially if you practice proper hygiene.
Assuming you're using an alcohol-based sanitizer, it's
plenty effective against viruses. Personally, I prefer good old fashioned soap and water as long as they're accessible though. Dries my hands out less and just as effective.
Again, if it makes you feel safer to wear gloves, be my guest. But the benefits are small enough that it's not worth calling people out for not using them - or even recommending their use. It's far more important to focus on the higher-percentage prevention methods like vaccination and masks.
