I prefer to see the tile removed. The more flexible (aka "thin") vinyl products (less than 8mm thick) have a tendency to telegraph subfloor abnormalities. You can float anything you want over top the tile...just be prepared to have MAJOR disappointment in the visual look OR move to a very thick, rigid, high-end vinyl product to get this to work. As I've always said (as my previous incarnation as Cancork), it is always better to remove old tile. Start fresh = a fresh start.

Sticky glue must be mechanically abraded off of the slab. No chemicals or steaming to force it further into the slab. Then the slab should be primed, leveled, and it’s ready to go. Glue can come up through the layers of the vinyl if it isn’t removed. Thinset is porous, as is wood underlayment.

Thanks so much for the detailed advice, @SJ McCarthy! I do have 3 extra boxes and then some, and so far pulling them up isn't seeming too difficult to do but I'm sure we'll run into some issues :-) The condo rules aren't very specific: 80% of floors must be carpet or area rugs - we have carpet in the bedrooms and someone on the board told me that LVP with an underlay is fine; this one has a pre-attached pad. But thanks for brining it up - I do want to run it by them once more to be sure before we totally finalize things. When you say "a little grinding PLUS a little leveling compound" are you referring to self leveling compound? It does seem to me that SL would be the best way to easily get a good level. We've worked with some patching material (Henry 345) on a few spots, and it has the appeal of potentially being simpler than SL up front (doesn't have to be mixed and doesn't require primer), but it's more expensive for covering a given area, and doesn't level itself. I'm still trying to figure out if we can reasonably apply self leveler to just a portion of the room, and I'm thinking mainly that will require that there is indeed a consistent "low" spot / sink so that the SL doesn't run out into the areas we don't want to repair. That does seem to be the case, but I need to really spend the time with a level to fully map it out. Thanks again!

Remove the tile. Sorry but vinyl needs PERFECT and FLAT. It is SUPER UNHAPPY when it is laid over 'wonky' and 'roller coaster'. The ONLY time you do NOT remove the tile: when you are a renter looking to put down (at your own expense) a TEMPORARY floor that is easier on your back/legs. And you will REMOVE your temporary fix prior to moving out. Believe me when I say this, you will be glad you did. We've seen people who tried this...and they were SPITTING mad when their floor bounced and clicked and squeaked and the edges broke and the floor 'split apart', etc, etc, etc. Sorry but disregard the 'hype' surrounding the sales pitch. The 'can go over top of any floor' is BS. The requirement for each flooring type is the same: subfloor must be smooth, flat and free of excess deflection and well adhered to the substrate. Is there a reason why you want to put down a $3/sf floor over top of a $20/sf floor?