But, that the hero's spirit doesn't exist in the AT is also an assumption, in fact, most proof goes against it, and the only solid ground given to the argument (that HoT!Link took the spirit with him when he traveled back in time) can easily be circumvented, as it's stated (by the King of Red Lions) in Wind Waker that he "lost the elements that made him a hero" when he traveled in time.
Further, HoW!Link being referred to as the "Hero of Time reborn" (by Ganondorf, of all people) being worthy of the ToC and the Master Sword, as well as being one of the few full Hylians (pointy eared people) rather than human and having physical resemblance to the HoT on top of it, it's almost spelling out a relation to him.
Likewise, we're shown multiple times that only the chosen hero stands a real fighting chance against whatever is going around, where the whole army of Hyrule and the sages combined were unable to defeat Ganon in the Imprisoning War, for one (also spelling a family-unfriendly teaching, in the Zelda series at least, unless you have special pedigree and divine help, don't expect to play hero anytime) so it's safe to assume that HoW!Link isn't just "a regular kid who turned into a hero"