I guess it depends what's under the cement under the tiles. If it were me, and I wanted to leave the existing floor as intact as possible, I'd make a solid base whose external dimensions were the same as the internal dimensions of your cabinet. I'd secure that to the floor with sufficient, suitable screws, first having shimmed it level. That would give you a robust start. I might even mount some L-brackets on that base in order to give added rigidity to the back of the cabinet. I'd then drop the cabinet over the base and secure it, probably with horizontal screws. Now your cabinet is attached to a virtually immovable platform, and possibly even reinforced with L-brackets. You'd cover the horizontal screws with the kick-plate trim on the base of the cabinet. It's not that big a job to drill clearance holes through the tile and/or hardwood and then insert suitable screw anchors into the cement (
if that's sufficiently thick/durable). An additional benefit is that using a solid base should spread the force of the screws and prevent tiles from cracking. Your local hardware store will advise you about the right fixings.
That said, you just can't beat a one-piece top (as in your latest image
Cabinets removed) as a way to give the cabinet maximum rigidity.
Looks as if you'll have to give up
something and moving the stove puts it into a much better position. So, losing the cabinet puts the stove in a better place
and removes your anxieties about stability and inadvertent demolition
Of course, you could leave things as they are. Just because you've had an idea, that doesn't make it the best course to adopt, upon mature reflection.
Or, have you considered moving?
