The one negative I could see w/ installing on top of the tile is that the tile surface can vary from tile to tile. It's easy when laying tile to have a 1/16" variance in how much grout is laid down or how hard the tile is pressed into the grout. But even so, I'd put the cabinets on the tile in a heartbeat.
Cabinets should be shimmed to level, so there is no need to fret over imperfections in the floor surface. The floor will ALWAYS be imperfect when it comes to preparing a level top surface of the cabinets to mate with an actually flat countertop.
In ye-olden times, people used wooden shims, which have the known potential to shink and move. Plastic shims are the least I'd consider putting under cabinets, but ever since I did an install with leveling feet, I'm NEVER going back to shims. Since I was only installing drawer bottoms, I went cheap and used top adjustment feet, which required drilling holes in the cabinet floor for allen key access (I then plugged the holes). Front adjustment feet cost a little more, but are more installer friendly, and don't require holes that might show (if the cabinet didn't have a drawer that covered them).
In an ideal world, with a kitchen over a concrete slab, I'd suggest having nothing but a vapor barrier under the cabinets, using metal adjustable feet to raise the cabinet tops to 34.5" AFF. I would strongly suggest continuing the flooring under the fridge and range, and either flooring, or in the very least shimming under the dishwasher, with a 3-sided drain pain under the DW to direct any leaks forward, where they will be seen.
For the OP, I would first consider whether the cabinets or flooring will stay longer. If the flooring is the answer, then just go over it. If you think you may replace the flooring before the cabinets, then mark and cut the floor along the front edge of where the cabinet bottoms sit, and the toe kick will hide this.
As for placing cabinets over gaps in the floor, just use leveling feet to raise the cabinets up. Don't bother extending the floor or shimming the surface. It makes no sense. Most feet will let you raise more than 2".