Kitchen is on its third diy remodel- first someone put tile there, and then shiplap, and now you want your use barn board. Is there by chance cement board screwed to the cabinet box or is this hackery directly on the cabinet? If there is cement board chip it off, if not I think you will be adding your barn wood over this mess as a surprise for the next person.

"adding your barn wood over this mess as a surprise for the next person".

Solutions are often so obvious they are funny.

I would also consider whether where this is in the kitchen would make this a good choice for barn board. Barn board isnt easily cleaned, and if this surface will get spills or grubby hands from kids or adults, choose a more easily cleaned material here and find a different spot for the barnboard.

I would not want reclaimed barnwood in or near my kitchen. It probably would be difficult to clean and it just gives your kitchen a very specific look. Can you post pix of your whole kitchen?

Can you use a 5-in-1 tool (example, example) to scrape away enough of the mastic to flatten the surface?