Can I restore my backsplash?

I would use the sanding flapper wheel on your drill motor. The wheel has a bunch of flaps of sand paper on it that does an amazing job removing paint. A narrow wire wheel on the mortar might be necessary to remove hard to reach paint but I would avoid hitting the brick with the wire wheel.

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Years ago we lived in Galveston, Texas in an old stilt house. It still had the original woodwork, but it was covered in numerous coats of paint to the point that the detailing couldn't be seen anymore. Hubby talked to the landlord and received permission to remove the paint and restain the wood to its original splendor. It took a while to do it, but he used a wire brush on his drill and was able to clean out all the layers without damaging the wood. You may be able to take the paint off with the wire brush on your drill without damaging the brick and grout, but you will have to be careful with how much pressure you put on it. It will be messy, but probably the best way to get rid of all the paint. If you can't do this, perhaps you can find paint the color of the brick and the grouting and do it that way.

Seems like it would be more trouble than its worth if its just that brick paneling. I would yank it off and get a new piece.,, https://www.homedepot.com/s/brick%2520panelling?NCNI-5

a more modern look might be to use tin ceiling tiles as a backsplash. https://www.homedepot.com/s/tin%2520ceiling%2520backsplash?NCNI-5

You can sand it but you'll need to seal it. It's porous so I doubt you'll ever get it all off. Food splatters and oil will be a problem on the original surface. What about picking a different paint color?

A lo of great tips. A loot of work, messy, and not all of it may come off. If you want the look of brick how about sponge painting using a few different color paints to get the look. Less trouble, less time, and less mess.

Then using a scrub brush with a back-and-forth motion, continue to scrub the brick to remove paint that has set deep into the pores of the brick.