I won't be able to offer suggestions, but folks are going to want to see pictures before they can really help you.

Yes...pls post photos of he entire space.

“Is it reasonable to not even have one at all?”

No. The purpose of a backsplash is to protect your cabinetry and drywall from spills. Spills can occur even on runs that don’t have sinks. If you want the protection without any “lines” consider having a 2” or 4” bs made of your counter material, using backpainted sheet glass, or using solid surface (i.e. Corian). There is a brilliant GW/Houzzer who had soapstone counters and used solid white Corian as her backsplash. It was gorgeous. In your space you could possibly continue the brick as well. Pictures would help.

Unfortunately, I cannot do pictures at this time. I appreciate the suggestion and advice about the necessity of a backsplash. A thought I did have was to use a large format tile. If we were to install a large enough tile so that there would be no horizontal grout lines, would that make sense? It would have a more solid surface look, especially if we kept a very narrow grout line with a matching grout. Thoughts?

What is your counter material?

With large format tiles, you have to make sure it doesn’t look like you put floor tiles on the backsplash.

There are people who‘ve used very scrubbable paint as their backsplash. Wouldn’t be my choice, but it could work for you.

No pics, no good advice.

Our counter is leathered Absolute Black granite.

Backsplash are not necessary; unrelated but In France, many modern kitchens don’t have backsplash - and when they do they often have a very low one (that do not go up to the cabinets). I can send you a link with pics if you would like. This also ables you to paint your back wall in a fun color: my mom had done red. That being said 20 years later the paint is chipping off a bit and we need to repaint. But we never had a an issue cleaning.