So I after extensive messing around with Perkons, I realize I had it all wrong.
Like many have said, it really is an instrument…it’s great for happy accidents, is very performable, and lends itself to live creation. But my problem was approaching it as a drum machine rather than a synth that makes a variety of crazy sounds.
The drum machine approach did not work for me at all…I just wasn’t crazy about the sounds, and the sequencer (while being pretty cool) didn’t jive with me for drum programming.
My own breakthrough was in abandoning a.) any pretense that I’d be using the Perkons for drums (other that weird perc sounds) and instead using a separate drum track, and b.) not trying to overthink things from a sound design perspective - and instead let the machine guide me.
In short, I was mid-curving it:

To put this in more concrete terms, I’ll have an underlying drum part and then just start tweaking everything on Perkons. If a sound works, I’ll blend it more heavily into the mix and start modifying it. If it doesn’t work, I’ll leave it out.
This is so liberating; I don’t have to to worry about returning to “that perfect closed hat” sound I had, or finding it in the first place…it’s just very organic. Along those lines, it’s also been helpful to generally just focus on 1-3 tracks at a time; with this approach, using all four simultaneously can get cluttered in the mix. What’s cool, though, is that this allows you to create some fun call-and-response jams where you alternate which parts are muted.
It’s also fun to save a pattern that’s a good jam, wreak absolute havoc on said pattern, and then revert to the saved pattern. (It would be cool if Erica added a Pattern Revert shortcut to do this without having to hit “pattern” and select the saved one). A randomizer for both the sequencer and kits would also be interesting to mess with.
What I’m left with, then, is a machine that is great for a variety of atonal fuckery, zappy sounds, weird textures, and so on. I find myself watching my hands randomly fly all over the machine, tweaking knobs, switches, and step buttons without giving it much thought at all. Live sound design without overthinking it! (My left brain will kick in when it’s time to add more variety, say, by adding a probability or odds trigger).
I’m still not sure if the sounds I get from this approach warrants inclusion in my live setup (I mean it is a really big freakin machine), but I’m finally having FUN with Perkons. My goal is to do most things live and without too much prep (a la Surgeon’s live sets), and I think I see a path forward here. Thanks to @DaveMech and others here for all the tips and encouraging advice. It really is a strange and interesting beast.