ASHRAE standards agree with you. Temperature difference in the room ought not be more than a degree from floor to chest height.
big job, in terms of time. Rip everything out. Put foam on floor slab to separate cold concrete below from what is above it. Put electric heat cables on top. Tile on that. Floor always warm.
The thinnest and best foam treated to accept tile on top is Wedi. 1/4", 1/2". 1" etc. The best heat cable company is Warmly Yours, in terms of being able to explain things to the general public and to provide service. Other companies can create problems or confusion. Always go back to Warmly Yours for answers.
second option : put a sound insulation membrane under wood floor. The kind of synthetic "felt" that goes under engineered hardwood. Or put foam underneath. then heat cables, then wood floor.
third option: put any membrane, anything, on the floor under whatever you put on top.