To your first question - Are HDTVs different than CRTs in this regard? YES. The CRT displays what the system spits out natural 1:1, it won't lag. Any HDTV will have some form of processing time, if not from stupid built in filters, but from the fact it needs to take that 240p signal and upscale it to a level the TV can output. Each compamy, each model(which vary internally aside from panel size) can process the data differently and as it does this, it adds input lag because the more it needs to chew on it to spit it out so it can be displayed allows the game to run out of sync X amount of milliseconds which for timing pissy old CRT games can be a problem. displaylag.com will explain ti better and show TVs where they're low enough (30ms and below) where you won't feel it. Mine I researched, it's at 25ms, and I can handle those, Punchout, shooters like Gradius, etc just fine. That's why I said what I did and how I said it, IF you played the 10-15 on the same TV (which you didn't, you used a CRT.) Plug that old console into the HDTV through the RCA jack, watch it lag out too as you described it. Go into your TV, see if the HDTV has a GAME MODE or if you can manually disable every bit of processing (filters, etc) it does to iron out visuals for movie watching and the sort and it should improve. My current TV you can disable every single filter which is why it can get by just through GAME MODE, but my older one I had to manually disable all the stuff on the Panasonic Viera I had (current is a Vizio) and it was fine.
I think the Genesis games look less blocky because they don't display graphics the same way Nintendo does theirs, and they also lack the color and added filters the SNES has which I think would play a factor into them looking less chunky.
DKC (and other ACM graphics games DKC2+3, Killer Instinct) would all look like crap. They're using a higher color table than most SNES games and using display tricks to get the color quality and smoothness up to where Nintendo could back in the mid 90s gloat they could keep up with 32bit systems visually. By using the 2xsai filter or whichever you are would take those very finely tuned and blended colors and visuals and just muddy them up badly. It's kind of like when someone used AA overkill on a N64 game ported from PSX and it would look splotchy or muddy and get dinged in online reviews over it.
Try something out for yourself and the others here since the thread is active. Take your NES, pop it into your HDTV, play Castlevania and Ninja Gaiden again, see if you have problems.