Tile – Sealing things under a furniture-style vanity

Back Story:

We are redoing our half-bath and are replacing a pedestal-style sink (the $75 ones you can find at Lowe's) with a much nicer furniture-style vanity (specifically this vanity). So we emptied the room, replaced electrical and plumbing (where appropriate), painted, and are now going to put the new vanity back in so we can finish the plumbing.

Our problem:

I had miscalculated the clearance needed for the "bear-claw feet" of the furniture-style vanity against the wall and the trim didn't let the vanity sit flush against the wall. So to fix this, I was able to cut away a couple pieces of quarter-round and drill some ~3/4" holes into the rest of the trim (to fit the contour of the bear-claw feet) and this now allows me to get the vanity to sit flush against the wall. However, in doing all of this, I have exposed some gaps from how the floor has been tiled. So my question is: How should I seal these gaps? Given that this is beneath the furniture-style vanity with an open bottom (please see pictures from the link above), this seems like a very important thing to do (our furnace is right below this bathroom so water leaking down would be very bad!).

Should I just slap down a bunch of caulk and call it a day? Given that I'll NEVER be able to re-caulk this in the future (while it is open, it's not open enough to access for maintenance), this seems like it would be a problem in the future.

More background on how our tile is laid:

So the bathroom's trim goes down to the baseboard on the drywall, then the bathroom was tiled with a ~0.25" gap between the edge of the tile and the trim. Next, a piece of quarter-round bridges that gap all around the room except where I had to cut the two pieces off. Again, right below this bathroom is our furnace so I really want to make sure we don't have a $10k problem if a sink overflows!