I can comment on my step son's opinion of HBK. His name is Jacob, and he's currently 10 years old (born in 2001). He started watching wrestling around 2006/07, and he LOVED DX when they had their resurgence...I can't remember if he was watching when they feuded with The Spirit Squad and all that, but I do remember that he mostly favored HBK over HHH.
Obviously Jacob also loved all the "kid-friendly" superstars; Jeff Hardy, Rey Mysterio, John Cena & CM Punk (of all people). Well, he loved CM Punk up until he turned heel by cashing in his MitB case on Hardy. He pretty much had a typical "young" view of the product; hating ALL of the heels, and liking almost every babyface. Come to think of it, I think that the main reason he loved CM Punk when he first started watching is because CM Punk was the World Heavyweight Champ at the time (obviously that was when Punk was still a face, before the Straight Edge Society). He also HATED Edge at that time. Man, there was sooo much passion in that kid's brain for Edge! LOL!!
Quick sidenote: Jacob also thought that any "older" wrestler who wasn't currently on the WWE roster (ie: The Rock, Hogan, Stone Cold, etc) was dead! Haha! Any time the announcers would mention Hogan (or any "older" wrestler) Jacob would turn to me and say "...isn't that guy dead?". Just thought I'd mention it, since it adds to the perspective of such a young fan.
Jacob also told me in the last couple of years that his favorite tag teams of all time were The Hardy Boyz & DX. He only saw the Hardys wrestle as a tag team a handful of times (unless you count DVDs, etc), and I think he only liked Matt because of his association with Jeff (who is probably his favorite wrestler to this day, even though he doesn't care to watch Impact with me). Obviously Jacob only thinks of DX as HBK & HHH, since he was too young to see the group's original debut or the DX Army (with X-Pac & The NAO). I never heard Jacob make any mention of HBK (or HHH, for that matter) before the last DX "reunion". I know the last incarnation of DX was "cheesy" (like BizzarroNavarro already mentioned), but I think it was a fantastic way to get HBK & HHH into the minds of the younger fans. Obviously it did wonders for Jacob (& his friends as well).
Jacob hasn't been watching as much wrestling for the last couple of years. I think the main reason is because his friends don't like it as much, and he just hasn't cared about it. He stopped watching wrestling soon after DX "split up" again...I can't even remember how HBK & HHH stopped tagging together again, but the last angle I can remember Jacob really going ape-shit over was when HHH went to Orton's house with the sledge hammer (I'm pretty sure that was after the new DX had split).
If most children share the same opinion as my kid, then they must have also loved HBK's character while he was tagging with HHH as "DX". It seems that Jacob stopped caring about him as much around the time Hornswoggle became an "honorary" member of DX. Although Jacob always preferred watching the actual wrestling over the "talking" (promos, interviews, etc), he was always most interested in spot-fest matches (ladder matches, hardcore, etc). I'd have to assume that most of the younger viewers these days are more into seeing a fast-paced match with a lot of high spots as well. Unfortunately, I don't think most kids appreciate the style that HBK wrestles/wrestled. Although HBK's a true artist in the ring (the story-telling, pacing, all of it), most kids today just want to see Jeff Hardy jump off a ladder.