Damaged wood kitchen cabinet repair

I recently replaced the countertops in my kitchen. The previous owners had a solid surface counter top. When the installers came to install my new quartz, they pulled up the old solid surface including the 4 inch backsplash of the same material. Unfortunately, the previous owners had also attached this backsplash to the side of a wood cabinet that faced a little coffee nook. When they pulled up the backsplash, it left some silicone caulk residue (which I have been attempting to remove with Goo gone caulk remover) and pulled up part of the wood. I am in need of some advice on how to fix this. The other option is to put up a 4 inch backsplash of the same quartz counter I had installed but I am really trying to avoid that. ANY advice is very appreciated! Thank you!

 

Silicone caulk cannot be dissolved.
What you are doing is using the Goo Gone as an oil to get between the silicone and the "wood".

Sometimes you are better off using your finger to slide across the silicone to try to get it to peel. Using metal tools will not work. You could try a plastic scraper.

I can't tell what that cabinet is made of so it's kind of hard suggesting a repair.
It looks like a paper or vinyl over maybe particle board. That would be very difficult to repair.

A close up of the damaged area may help.

 

I tear out counters quite a bit. I would advise you go with the side splash.
If your cabinet face frames have a standard 1/4" overhang at the sides, you could stain a new panel to cover the side. We call it an end panel or skinning.