I think the paint would make it a bit less slick, but painting tile in a entry/foyer area is probably a bad idea because even if you prime and paint it perfectly, it probably won't hold up or be durable enough. I think painting tile is okay, but not for areas that you walk on.

Just rip it out and replace it with a less slippery tile. If it's a small space' its just a dirty DIY job, not a difficult one. Painting is a 2 day job for a 10 minute result.

Thanks paintguy and live_wire_oak! The prime/paint/seal process does indeed look like a many-day process. I'm encouraged that you think it wouldn't be hard to tear out the old tile and replace it. I had been thinking that wouldn't be practical.

Maybe cover it with a nice looking cut to fit all weather rug?

Two thoughts: Of course, have two carpets--one little all-weather/washable at the point when those snowy boots hit your foyer. And another one a bit further, large enough to cover most of your foyer. You can find marvelous Oriental Rugs, either handmade or machine made that will do the job. And, those snowy boots should not get further in your house than carpet number one.