You can remove the rest of the grout in the area where it is loose from the high pressure cleaning and regrout that area only. I doubt that the tile is over hardwood since it doesn't sound like it's breaking up in a lot of areas and it's been down for at least 5 years. I would suggest getting a tile guy to take a look at it and give you his recommendations. Make sure it's someone who is a real pro who has referrences that you can count on. Talk to friends and neighbors, check with a couple of decorators in your area to get names of tile setters that come highly recommended with years in the business.

Actually looks pretty nice, but I understand your wanting to change things around. Option one would be to pull the grout and change the colour. Option two is to tile over. The only worry here is twofold, There is the height differential you mentioned to deal with. You can mechanically "scarify" the existing tile and tile over it...and that's a dusty "pain." Also, you are adding considerable weight to the floor, which COULD cause problems down the road if the deflection ratio is excessive. Also, you are relying in the integrity of the installation of the underlying tile and how well it was installed. In other words, it will only be as good as the adhesion of the original tile job. Frankly, I would bite the bullet and rip out the old floor and install whatever it is you want there. Dusty and messy? Sure.....but it's what a Pro would do.

Be sure to get a 2nd opinion! I bought a house in horrible shape... peeking under the (icky!) carpets, I could see hardwood floors. Once the carpets were removed, there was white and black 'crap' stuck all over it. The first guy that looked at it told me it could NOT be removed by machine/sanding because it would gum up everything. He said to remove it by hand. I scraped one day until my arm felt like a giant lobster's claw and had only managed to remove a few square feet. I found another guy to come look at it and he said: "no big deal! my machine will take that off in no time!". I hired him. :) My LR, DR, and bedrooms also have 2.25" oak flooring. There was severe damage and buckling of the DR floor in one section due to water damage from an unattended leaking roof. The 2nd guy told me it would be cheaper to fix it myself (rather than have him do it) and he suggested taking wood from a closet. Luckily, a friend noticed that the hardwood in the closet was 3" wide! so be sure to check that. lol! On the damaged section, he told me to chisel out the bad section... I couldn't find the tongue-n-groove oak in 2.25" to replace it, so he sent me to a hardwood mill to have custom pieces cut. My repairs came out great -- thanks to him. He said my only mistake was that I should have ordered "2nd" quality oak so that it would better match the existing older wood. i.e., my fix was 'too perfect'. :) I was bad about taking any 'before' photos (it was all just too ugly!) -- but this shows the buckets collecting the dripping water before the carpet was ripped up. That white-ish and blue crap around the carpet edge was just some of what was stuck to the floor. And here's an 'after' photo (with much more work to be done!) from a slightly different angle: PS: yeah... the odd-shaped cut-out in the wall was changed (but remains, despite my father's recommendation) because the kitchen cupboards had to be lowered for me to reach them. That cut-out gives me a *perfect* view of everything from the kitchen, and makes a great 'pass through'. :)

Do nothing...and install the LVP as-is. The terrazzo you have is a HIGH VALUE floor. I mean SUPER expensive to duplicate. I would 'give up' on the warranty of an LVP floor and just toss it in. To do anything to this floor will ruin it for future generations. To level it you would have to grind, grind, grind, grind to get SLC to grab. Once you do that, you have lost the value of the item. At that point, you might as well spend the $3/sf to jack this out. It will be worthless. The TILE is see in the background can be left in place, you can do a bit of grinding and then fill the grout lines with SLC. You need to do some scarification of the tile to get the SLC to GRAB the tile. So long as it grabs you can fill the grout lines and flatten it to the best of your ability. I would EXPECT to use T-moldings where the two floors meet. The only way to avoid this would be to remove it all. In essence you are covering the HIGH VALUE products with a lower-value product. I get it. The current floors are not very attractive (compared to today's trends). But realize the value of the terrazzo. If you were to replicate the floor today (with modernized product) you would be looking at $25-$30/sf. Compare that to the $4/sf product you are going to cover it with. Right. LVP is a downgrade compared to terrazzo. No way around it. I would leave the terrazzo in tact and untouched. I would give up on any warranty and install the LVP over top. The tile can be dealt with as you wish. But remember to put in t-moldings between the LVP sitting over the terrazzo and the LVP sitting over the tile. NO underlayment. Most LVP's do NOT allow underlayment. Especially if they have attached underpad. This is LVP not laminate. If you want to get a cork underlay on these floors to help even out the surface then work with $4/sf laminate. That will buy you a TOP OF THE LINE European laminate. You can then use 6mm cork to correct some of the unevenness. But LVP will not allow that type of underpad.