I'm moving to mesh shoes too. Gaitors could be good for snow when I'm wearing a gore-tex sock, but that assumes the gaitor is high enough to cover the opening to the gore-tex sock.
Otherwise, it's about keeping things out of the shoe. But you're right, some of that stuff is going to come in via the mesh. I guess it's about keeping little rocks and sticks out that are of a particular size, bigger than the mesh allows, but small enough to work in and get into the shoe.
Thanks --- this is a great "challenging" question. Now I'm wondering if this isn't a little like trekking pole baskets, where I've concluded that I'll either bring a real (separately purchased) snow basket, or just leave them off entirely. I'll try skipping the gaitors on my next trip (next month) and see how I like that. I certainly like the "one less darned thing to deal with" aspect.
I guess one other reason is to keep your upper shoes and socks clean, but ... that's a little weird, given that the lower socks will get filthy as might your leg above the gaitor.
Maybe it's just a matter of the terrain you're in, i.e., some locales might offer a lot more things that get into the shoe and bother you than others do.