That is exactly what we are concerned about. The new sink should be here in a day or so. We are hoping it will be a bit bigger than the old one but if it is smaller we will have a gap. If that happens we may need to find another piece of gumwood and re-make the front panel-- we really do not want to have to do this. But we recognize we may have to. We are getting this gumwood from the center panels of old doors that we buy at salvage. We learned the hard way that it is the only way to match the old gumwood that is already in the kitchen. The front panel that is under the sink is contiguous with the two stiles that run down on either side of the two doors in the cabinet below the sink. My DH took such pains to craft it and we are sick about the possibility of it not fitting the new sink.

I'm normally very picky/neurotic about things like that, but I have to say, based on your photo, that a good caulking job should make that disappear, and the caulking will hardly be noticeable. You may keep looking at it and be bothered for a couple of weeks afterwards...and then one day you'll realize that you don't notice it anymore and no one else does either. I would venture that, even if the cutouts had been better executed, you would have needed some caulking anyway to hide the inevitable slight gap, and to prevent water getting in between the wood and the side of the sink. Finally, do you really want the cabinet to be re-made and have further delay over this issue?

"Is it "normal" to cut away that much of the sides of the cabinet to accommodate a sink? Yes. "Can a new sink be mounted to quartz?" Yes. "Do the sides of the cabinets need to be built back up first?" No. Not if you strap and undermount a sink in place. (Hercules Universal Sink Harness from Braxton Bragg or Sink Strap from Regent Stone Products). "Can this current cabinet even support the weight of quartz + a sink + water/pots in a new sink?" Yes, when you strap a sink in place, you are creating a truss. You are replacing the cabinet sides with 9" of stainless steel or Silegranit. I'm undermounting a 22" x 33" Kohler Vault into a granite top on a 27" cabinet for a customer as soon as the sink arrives. I put a 33 1/2" Silegranit into a 33" cabinet last week: Note the top of the cabinet sides have been removed to make way for the sink flange. Those are Sink Straps pictured. The cabinet front will be reinstalled to keep tension on the straps. Minimal clearance between cabinet wall and sink. The installer doesn't need access to tighten the sink in place.

I replaced a 33" drop in double bowl sink with a 33" single bowl sink. After install, I saw that I could easily have gone with a 30" sink. The 33" is huge! Gained so much space by going with stainless steel vs the old thick cast iron and losing the divider. OTOH, I needed the 33" to fill the hole left from removing the old 33" sink. But I made a note for next time when I am replacing counters. One other thing/ question. You said you needed to cut into a blind cabinet. Is this blind cabinet there because it is a corner? Running a sink past a corner is going to look weird and perhaps not be quite so functional as you are maybe thinking.