Well, you're probably not going to get a strait answer about joysticks.
Happ Comps are about as the American Style as it gets.
They are probably identical to what you remember from 90's beat 'em ups.
There has never been any consensus on what the best "all around" stick is.
Certain sticks work better for certain genres of games.
You'll do better in shmups with an extremely short throw and engage distance. (Seimitsu LS-56, Suzo 500)
For fighters, you'll pull off special moves more consistently with a longer engage distance and equal engagement areas for all 8 directions. (Sanwa JLF)
For classics, you'll be better off using a physical 4-way restrictor.
The closest thing to an all-arounder that most people agree on is probably a Sanwa JLW with the restrictor set up so it can easily be changed between 4 and 8 way.
Ultimarc sells it under the name j-stick. Groovygamegear sells a similar one under the name Omni2, although I'm not sure if it is a genuine Sanwa or a knock-off.
The stock spring is probably too light to have the feel you are looking for though. IIRC, the spring Ultimarc sells for the U360 will work in it.
I have Sanwa JLFs in my personal cab. When I first got them, I described the feeling as "dainty". The spring force was just too light for my taste because I was used to American cabs. They've grown on me as I've heavily modified them with different balltops, springs, and optical switches. I also had custom shafts made to add rotary function, but haven't got around to making the rest of the parts. When all is said and done, I'll probably have $200 in them, but I enjoy the tinkering.
I have Happ Comps in the X-Arcade setup that my guests use. It's about the feel most people remember.
My personal opinion is that the Happ comps are going to be closer to what you remember when playing Street Fighter, Mortal Kombat, & beat 'em ups.
Are there joysticks out there that are going to hit the corners better and allow you to do the special moves more consistently?
Yes, and they will have that "soft touch feel" you said you didn't like.