There actually was someone who did a brick backsplash on GW. Have you checked through the FInished Kitchen Blog and the Backsplash slie show? It would be helpful if we can find that one.
If you really like something than there is a way to accomplish it. It doesn't matter if it is popular at the moment.
Can you post a picture? Subway tiles are put in in a brick pattern. Lots and lots of colors to choose from.

For wear issues, I'd rather do a brick shaped tile than to paint tile. It will look neater as well. Find a nice ceramic or stone tile and get it in a long skinny shape like a brick and less like a subway....
Brick is also thick unless you use veneer, and can be more expensive. Painting it adds a bunch more work to your project as well.

I did a brick wall in my kitchen, but left it as is, I did not paint it. My brick is unsealed, I have no issues with cleaning it at all. HOWEVER you are specifically asking about painting it. In my opinion, I don't think I would do that. Although I always envisioned that some 10-15 years down the road if I get bored with my kitchen I could change it up with painting the brick, walls, cabinets, as people do. But I envisioned maybe doing a whitewash kinda thing, I think it's called limewash?? Or somehow giving it that crumbly/whiteish look.

Thin brick veneer comes in many different colors, not just the classic red brick. I'd look into that first, as opposed to painting it. Painting it - I think - will make it look a bit more one dimensional. The look of the brick with its "mortar" (grout) is what makes it look interesting, textural, and authentic.

Is it just the texture you are after?? If so brick gives you that, if it's just a matter of color matching your countertops, then you can install beadboard and paint that. Or just leave the walls as is and paint that. Paint is so scrubbable these days, and who cares if you'd have to touch up the paint in two years! But then I would do ANYTHING besides tile, but that's just me.

I love painted brick and was planning on doing a white painted brick (veneer) backsplash.

I liked the texture of the brick but wanted the clean look of the white paint (I have bathrooms filled with white tile and did not want it in the kitchen). I also felt like the brick had heft and substance, which was important to me. I think it is classic and unexpected all at the same time.

BTW, many people tried to talk me out of this "because it is just is not done these days" - which makes it all the better IMO.

I don't think cleaning should be a problem as long as you choose high quality paint - and DIY paint touch ups are easy (and cheap!).

I actually ended up going with a slab backsplash after falling hard for my quartzite. But if you do it, please post pics, so I can live vicariously... :-)

At first the idea of painted brick seems terrible.
BUT then, I found this picture and I realize my vision
is sooooo wrong. Brick painted is lovely.

I still like natural brick but I have a much nicer view
of painted brick.

Here is a natural brick. Yes, it is pretty.
~boxerpups