I'd vote to do it right the first time, which would be to rip up the tile and then replace flooring. However, if that's not feasible floating flooring is an option.

Here's the thing about floating flooring though. It's going to raise your floor level so you might run into issues with your cabinets and doors. Which means you might have to replace or alter those, which again is an added expense. Plus, it seems like your floor is uneven which is going to be an issue for almost any flooring.

I know it seems like a lot of money now, but you will be so happy if you rip up ALL the tile first, in both kitchen and living room. As mentioned above, if you lay flooring on top of that uneven tile, you may run in to more problems down the line, costing more $$$ and headaches. If you rip up the tile, you can then still do an affordable laminate or LVP, which is not expensive.

Doing it right will add value to your home, rather than problems and headaches, for both you and potential buyers down the road.

The absolute right way is to leave a functioning floor in place and float over it. Thatis the genius of the new technology that we have used since 1996... oh, 1996; I guess that is tried and true. The latest and best entry into this category appears to be rigid vinyl planks that are waterproof and look fantastic. They float over existing floors [you may beed to level areas with extreme lippage or fill extra wide grout lines like in sautillo tile floors)}. We just installed some in a bathroom, fbut not over existing floor. The problem with installing over existing floor is door thresholds to the exterior, or other rooms, toilet flanges and other issues that arise with raising your floor. Get a sample at home depot or floor n decor and lay it up to your thresholds to see if your doors will still work...you can cut doors down. Good luck.

"With stiffer floating floors like Ceramic, Laminate and Hardwood, it could lead to creaking noises and cracking and breaking of the locking mechanism. If the floor is rough and bumpy, it will need to be smoothed or skimmed out. Going over a rough subfloor will lead to uneven wear patterns even in a floating floor. The Clic together LVT products on the market today, require a smoother flatter floor, or they are very difficult to lock together. "

This article explains other potential problems.

https://www.google.com/amp/s/flooring-professionals.com/flooring-resources/why-floating-floors-fail/amp/

There’s only one right way. There are a lot more hack ways. 4K is cheap for that.

You would be surprised how fast it goes with the right tool from the rental shop

One of the biggest wastes of time and effort is in removing sound tile. There are few good reasons to do so, especially if you're covering it with a floating floor.

I had a contractor friend loose his shirt demolishing perfectly sound tile in a high rise condo years ago. Took forever and he nearly damaged the concrete structure. Scared me to death.

Get an estimate for a large piece of 12 foot wide wall to wall carpet that is cut to fit and edges finished to fit your living room space. The bit of tile you'll have around the edges won't be a big deal. You'll need a pad. This custom rug will serve very well for another year or two until you are ready to spend the money on a new floor.

Those are ceramic tiles over a concrete slab. Looking at the quality of the tiles I am going to say a DIY or lower end instal. I am going to bet they weren't back buttered and the thinset not burned into the cement.

What does this mean to you? They will pop right off. Go on amazon and buy yourself a Bosch hammer drill for about $250. They have a bit for tie removal. You will have that floor up in time for lunch.

If not look on Craigslist for demo specific workers. That should be $500 at most. Your flooring person is lazy move on and find someone else.

The problem with floating anything over this tile is the "uneveness" of one tile to another. So.....you need to KNOCK DOWN the high edges = sanding/grinding machine....several HUNDREDS of dollars for the rental.

Now that you have sanded the b-jasus out of your floors (you have to drape your walls because it is DUSTY AS ALL GET OUT) and you have been vacuuming up tile dust for weeks, you now need to FILL THE GROUT LINES.

Yes. You still need FLAT. So....fill the grout lines. Good. Now that you have that, you are ready for a floating floor.

Or....if you want to add Self Leveling Compound (SLC) you will need to STRIP THE TILE of any cleaning products (rental machine or professional cleaning company). Then you have to ROUGH UP the tile (sanding machine) so that the old surface has some GRIT to it. Vacuum up tile dust for several weeks.

Continue with PRIMER over the tile....so that the SLC can GRAB the tile. Let cure. Now you can apply 700sf of SLC (which is NOT self levelling BTW) over the entire surface of the tile.

Awesome. Now you add cork underlay (3mm is "ok" but 1/4" is better) and then install 10mm laminate (this is the lowest level of laminate I would look at). Excellent. Now that you have done all of that, your doors need to be rehung.

The 1/4" gap isn't enough for a floor that will sit 5/8" higher. Oh dear. That's several hundreds of dollars more just for the doors. And that's if you can trim them. If you can't, you need to buy new doors. Ouch.

The $4000 quote to remove the tile means it is $5.70/sf for the job. It sounds like you are in an EXPENSIVE area of the USA. The going rate in my area is a little bit more than $2/sf. Over slab it might be a bit closer to $3/sf to remove and dispose of the tile. Does that quote include grinding away the thin-set/mortar and "floor prep" to get the floor ready for a floating floor?

If it INCLUDES all of that, then the $5.70/sf is BANG ON. If you estimate $2.50/sf for tile removal and $3/sf for subfloor prep (includes some grinding and lots of SLC), then you are right on the $4000 bid you have been given.

Trust me. This is decent if it includes all of the prep as I described. And the cost to "go over" the tile with SLC = $4000. It is the same price. And then you get to ADD the cost of the DOORS because your new floor will sit HIGHER than 1/4"....I guarantee it.

The good news: once those tiles are GONE...you NEVER have to deal with them again. Ever. And your counter tops will remain "appropriate height" because you will maintain the distance from floor to counters.

My wife shoveled our tile kitchen floor out in one quarter the time it took me to remove two from the bathroom which had been properly repaired. Mine took plenty of concrete with them; hers none. Don’t ever believe all the tile are coming out as easily as those tested or bid. If your tile are sound and you can skip this risk, I’d suggest you do so.

Ouch!!! I'm having the same issue with a floor I have hated for 20 years. Best wishes. I just wanted to commiserate.

So the bid did include grinding thin set so that your slab would be ready for installing a new floor. That's good. So $2000 for tile removal and $2000 for grinding/subfloor prep. Area = 700sf. That is pretty much the going rate where I am...I'm in Canada where the cost of living is high and minimum wage is high....and trades people are in HIGH demand with VERY FEW trades people working = HIGH WAGES for trades (Red Seal Plumbers and Electricians bill out at $180/HOUR in my province...yes...they make more per hour than a family physician!).