You don't want a sea of sameness where there are no demarcations between surfaces. You need some form of contrast between surfaces for visual clues as to depth of field.

Stairs should match the floors. Period. You need new bamboo treads.

luvbeautifulthings: I live in SoCal, where houses cost what yours cost. I have stated before in these forums that I do see LVP in my area. But we live in a suburban area of large families, SAH moms, pets--that kind of neighborhood. It's not West L.A., if you know what I mean. I agree it's important what realtors tell you about the local market. In your case, "floating" the floor on top is going to mean a couple of things you might not like; such as having to cut the door molding around every door casing. Removing and re-installing the baseboards. Will your doors still shut or do those have to be cut down too? How will the new floors transition to the kitchen? (Ugly transition strips?) Or are you taking LVP right into the kitchen...in which case you're going to (maybe) be fencing in your appliances (How will you now slide your dishwasher out?). The other thing is this: as you search for your awesome LVP, that looks realistic, that's good to excellent quality, you're going to find you're almost spending as much as engineered wood. And as we both know, the cost of installation in SoCal is high--- product cost isn't going to vary all that much unless you go really bottom basement (but I have a feeling you won't). Near me, installation was $5-6/sf and that was 5 years ago (or so). It's probably higher now. The cost of glue for my downstairs (which is large) was $2,000. Not a budget breaker in the scheme of things.... so given the choice, glue down engineered works better where we live (we are on slab). Either way, you'll have to redo your baseboards. Price it out. I'd demo that floor and not think twice. I'd get the floor you want and do it right the first time. Glue it down.

OK...cork can be floated underneath a floating floor (like laminate or an engineered hardwood that is floated) or it can be GLUED DOWN (as in glued down underneath a glue down hardwood or underneath ceramic/porcelain/stone tiles). Cork should NEVER be pierced by metal. It will shred itself into cork dust over a few years. Agglomerated cork (stuff used as underlay or cork boards) is wonderful stuff (I'm a cork expert...so this is my wheelhouse) but it is expansive. If you pierce the cork with THOUSANDS of metal spikes (aka nails, cleats, staples) it will rub up against those spikes with the massive change of temperature in an unconditioned space such as your space. That rubbing means it will rub itself raw until it crumbles...and then you are in for some serious issues...It would be like nailing down hardwood over potato flakes. It would be hollow and rickity within 5-10 years. You would want to glue down the cork, then glue down the hardwood. But then you have to have a some form of conditioning because the glue then becomes the weakest link in the construction. Most indoor wood adhesives are temperature sensitive. They deteriorate if temps spike above 85F (they liquify) or if the temps drop below 35F (they become brittle and then crack and break). And just so you know, a floating hardwood has the same issues with the glue on the joints...the same type of glue is used on the edges to keep a floating hardwood locked together.