Render / paint etc... not so relevant in your case.
In general, though, for old lime mortar solid walls, anything other than lime can be problematic where water / moisture / damp is involved - Things that keep water out tend to also keep water in even more effectively (and while they may work perfectly when they are first applied, over time they will degrade and let water in).
If you go back to porous lime (which is what that gravelly filler crap looks like) it would be able to dry out. When you have the extraction in, and wait for a month or two over winter, you can see if that's the proximate cause if it dries out.
Unfortunately, a lot of fixing problems involve "try this one thing. wait a long time. didn't work? try something else".
So, to the floor, would a stone work top need to be adhered to the wall or could I get it fixed just to the unit. It's a single unit 800*300mm with upturns, just to make coffee on. Then I could have that unit mobile if I decided to be an adult and actually put the time into fixing it.
Render / paint etc... not so relevant in your case.
In general, though, for old lime mortar solid walls, anything other than lime can be problematic where water / moisture / damp is involved - Things that keep water out tend to also keep water in even more effectively (and while they may work perfectly when they are first applied, over time they will degrade and let water in).
If you go back to porous lime (which is what that gravelly filler crap looks like) it would be able to dry out. When you have the extraction in, and wait for a month or two over winter, you can see if that's the proximate cause if it dries out.
Unfortunately, a lot of fixing problems involve "try this one thing. wait a long time. didn't work? try something else".