Unless I'm completely misunderstanding your quest, your yearning reflects a fundamental failure of Korg's marketing department. In a bid to stay relevant as a keyboard workstation manufacturer while the whole world went inside the box, Korg took some weird sequencing stuff from this guy named Stephen Kay, raided their stillborn OASYS workstation (not that one, the one it stole the name from) and came out with the Karma, basically a much-maligned play-by-numbers workstation that attempted to bridge two worlds that hated each other - on-stage cover musicians and basement-dwelling Ableton nerds. It lasted not very long. Either you were exactly the guy that wanted standard corporate Korg workstation sounds combined with weird-ass Ableton tweakology or you hated the abomination. They were pretty much always available at steep discount from music stores. Eventually they came out with the actual OASYS and put "Karma technology" on it because bloody hell, they were proud of it and if you were still buying keyboard workstations in 2005 you sure needed all the onboard sequencer fun'n'games you could afford. Even now, sixteen years later, the Kronos and every other big keyboard workstation you've never considered has "Karma" on it. Thing is, though, you're still talking about mashing a button and launching a string which is far more in the realm of the Akai MPC school. See also: NI Maschine. "Classical groove boxes" means MPC60. You can download it for iPhone for like $10. My buddy Dante put this song together on his phone in like half an hour.my initial forays into searching the controller world suggest this isn't a thing.
Those Korg videos. Those came out post 90s? It does the kind of thing I want, but probably a little too much menu diving for my tastes. I think what I want is a bit more bare bones even: like the sequencer built into the volca, except like 3 of them stacked together, with some more buttons. Even the arp on the volca would be fine with a couple more functionalities and a few more buttons rather than all the settings jammed in 10 positions on a 1 cm diameter knob.
Honestly? 1) Stack sequencers 2) Buy some Doepfer 3) Learn Reaktor or Bidule, which is hardly stepping out-of-the-box. The problem is MIDI is a digital protocol which means the chipbenders don't fuck with it much, but what you're looking for is pretty advanced analog behavior which means you either need to be a diehard chipbender or you need to do it in the box. In the box is so much easier.