Our mudroom is like yours. The flooring is LVT, I also have a 10'x3' Waterhog mat, and a rubber boot tray to keep the wet/muddy shoes in. Kiddos are young and the flooring stays dry.

Yes, it would be crazy. Keeping the mudroom floor clean is a constant chore, even more so in the winter months, everything gets tracked in. Strongly suggest you stick with some sort of tile -- it is bullet-proof. I personally wouldn't suggest LVT based on my use of mudroom (direct entry from garage = constant dirt/dust/debris, and dog going in and out), but if you don't have a mudroom that's going to get a lot in terms of daily wear, tear, and dirt then the LVT is probably fine.

I would advise wood floors there, mostly because of all the wet exposure to rain, snow, etc. unless you cover it with a good size area rug to catch the moisture.

Is tile an option?

^^ Do you mean not advise wood?