Wood is hydroscopic. It gains and loses moisture. With that gain and loss, it shrinks and swells. Gaining or losing moisture from the airs humidity, takes time, it doesn't happen in a few days. It normally takes a couple of weeks to really fully come to moisture content equilibrium. Wood is being wood. Wood floors and subfloors, will squeak over time of seasonal humidity changes, unless they are fully glued down How long was the flooring allowed to acclimate to the interiors humidity? Was it verified by a hygrometer and a wood moisture meter?

You need to do a little more archeological digging still. You need to know the whole construction of the floor before you can decide how best to install something new. I'd take up that plywood piece in the kitchen and see what's under it. It's higher than the surrounding floor, so needs to go. You should also be able to see from there what subfloor is under the original kitchen hardwood. It's too bad the space under your entryway is finished, but I'd still do some exploring from below to see what's under there, and to figure out the fix to stop it from squeaking. Is it drywall below or a drop ceiling? I know it probably seems like a pain to tear out drywall, but if you cut out a clean patch, like 2' x the width of the joists, it'll be relatively simple to repair. Just make sure to cut it down the center of the joists so that there's room to screw the patch piece to it later. Cut the hole under the area that squeaks the most, if you can. Have someone walk on and watch the floor from below to see what's moving. It might be as simple as pounding in a couple shims from below, or face-nailing a loose board from above. You want to find out what's underneath the hardwood. It might be laid directly on the floor joists, or there might be 10" wide boards that the hardwood is nailed to. Another thought is to take out a board in the hallway, since that's not original and you want to replace it. See what the subfloor is there. I am a bit puzzled at why, when you've stripped the kitchen down to the original hardwood, it's still higher than the hallway floor. Is your current surface perhaps not the original kitchen hardwood? Peeling back the plywood area will help determine that. BTW, I haven't heard of that stop squeaking product you linked to, so can't offer any personal opinion. And don't pour self-leveling compound over top of hardwood. It needs to go over plywood.

I am thinking about using a shellac finish and sealer on this hardwood, which will be the floor for a kid's room. This Bullseye traditional finish and sealer product from Zinsser seems widely used and well liked: https://www.lowes.com/pd/Zinsser-Bulls-Eye-Clear-Shellac-Actual-Net-Contents-32-fl-oz/3449840 This particular product says it is clear, and I think there is an amber tinted version. Can shellac be applied without problems to hardwood? I believe I could put poly on top of the shellac and be done. Is this right? Any tips or warnings? There is also this Sealcoat by Zinsser, which does not appear to be shellac based. https://www.amazon.com/Rust-Oleum-Zinsser-854-Sealcoat-Universal/dp/B000C02BXW/ref=sr_1_3_mod_primary_new?dchild=1&keywords=zinsser+shellac+sealer&qid=1610393849&sbo=RZvfv%2F%2FHxDF%2BO5021pAnSA%3D%3D&sr=8-3 Would the Sealcoat be a better option to just put over the hardwood? I have read elsewhere that shellac can be applied over poly, which is good since I don't know what is on the existing floor. Other places say shellac could be sanded after application, so when we have a chance to completely sand and refinish this floor, I would probably want to do this. Does this sound like an acceptable temporary fix, or is there something I should consider before proceeding? Would shellac ruin or damage the hardwood. I am also thinking about LVP, but I am concerned this hardwood floor is not flat enough for that. I understand the subfloor for LVP must be 3/16" within about a 1 foot diameter to work. Any ideas about this? Is there a way eyeball this floor to determine whether click LVP could be put down? Thanks!

You would need to strip the polish off.