God Hand isn't sidescrolling! (And has a character progression system!)
But it's mad fun, so yeah, play it.
The whole "sidescrolling beat-'em-up with a level system" thing (not sidescrolling action RPGs as much; when I think those, I think post-SotN 2D
Castlevania, a very different sort of game) has been around since at least
1996, and even
River City Ransom on the NES had a stat system, so it's not exactly a recent trend. It stuck because people apparently like it more than
Final Fight-style "enemies get stronger and you don't" beat-'em-ups, and that's part of why I can't think of any modern sidescrolling beat-'em-ups off the top of my head that don't have some sort of character progression system.
Anyway, some of my favorite modern (past decade or so) sidescrolling beat-'em-ups, plus a few digital rereleases:
Viewtiful JoeOdin Sphere (your mileage may vary, but it's pretty cheap now that it's available digitally on PS3)
Guardian Heroes (available on XBLA for $10 with online play and stuff added)
Advance Guardian HeroesCode of PrincessFinal Fight: Double Impact (you already mentioned liking
Final Fight, but if you haven't played
Magic Sword, it's definitely worth a try)
Definitely play all of LD's suggestions, as well.
Scott Pilgrim and
Double Dragon Neon are both available for free* through
PlayStation Plus as part of the Instant Game Collection, so if you have a PS3 (or a friend with one and a PS+ subscription), you can get immediate access to both that way.
If you're willing to emulate stuff, try some mid-'90s and later arcade games like
Tower of Doom and
Shadow over Mystara. Definitely try for four-player co-op if you can;
Shadow over Mystara has a designated channel in the
GGPO client, so you may be able to get a game together there. You'll likely find hundreds of other arcade beat-'em-ups looking through a MAME list, as well, though whether or not most of them are any good is another story.
Unfortunately, the modern fork on the evolutionary path of beat-'em-ups, hack-and-slash-and-maybe-QTE-'em-up games like
Devil May Cry and
God of War, is even further from what you're asking for than anything I've listed. Definitely worth playing, though.