I think answers to this question are missing the part about “easily exporting into Ableton”
I have no experience with boxes that can do this but do know that suggestions for anything Elektron are a little off the mark - none of them can “export” - unless the OP is also OK with recording stems individually then that kind of opens up this thread into something very, very general “what’s a good groovebox”, no?
The MPC is just a sampler right? Not so interested in going around hunting for samples, but if there are decent sounding sample packs for it out there, might consider it!
Just want to say thank you to all of you who have responded so throughly and thoughtfully!
I really appreciate it; left me much to consider.
Also want to mention that it’s very refreshing to meet a bunch of kind strangers on the internet! Not something you see every day.
Thanks.
-Sebastian
Yeah if you like MC-101 sounds and its export to Ableton process, then MC-707 is probably a good fit for you. Unlike going for an MPC One or Elektron box, you won’t be looking at a whole new learning curve. MC-707 is not that big of a box, gives you access to all ZenCore synth parameters, decent workflow for creating clips, arranging clips into scenes, then lining up scenes into songs. MC-707 projects can be easily moved to your MC-101 by simply moving the SD card from the 707 to the 101 - provided you keep your projects at 4 tracks max (101 can play up to 4 tracks vs. 707’s 8 track max).
Push 3 Controller (not Standalone) would have been a good fit in terms of budget and Ableton integration, but OP implies a desire to work away from the computer.
I’ve talked about this before in other threads but I tried going from the MC-101 to the MC-707 and it felt like going backwards, despite the bigger screen and higher track count. The 707 just didn’t feel fun to use at all. I’ve also had an MPC One, which was fine for a bit but ultimately fell short for me, though many others here have had success with them.
I finally settled on the Analog Rytm (mk1) which I think strikes a good balance between being capable and still fun to use. You can connect it to Ableton via Overbridge. The Digitone I found more limited as a ‘groovebox’ given the track count limitations, but YMMV. The Syntakt is similar to the AR in many respects but I enjoy the form factor, the pads, the kit workflow, and the individual outs of the AR much more.
Maybe the best advice is to get a competent sequencer first, even if does not make any sound, and you control the daw first.
There are videos for any oxi combination, also with mc707, where the users report that this solves multiple problems with the device.
True for the ancient MPCs. Even the MPC 5000 had an onboard synth, which IMO sounds quite decent.
Since the “new” versions like MPC One, Live, and X the MPC includes a couple of different synths and sample instruments. It’s like a DAW with plug-ins in the box … 
It’s - as so often - depending on individual preferences. I got the MC-707 on my table and after some videos I was a bit sceptical, whether that UI and workflow would fit to me, but I thought that the sound engine and the clip concept was exactly what I was looking for. After some hours of getting familiarised, I loved it
Korg electribe 2 has ableton live set export built in the hardware. It’s only audio do, in a ableton live set with scenes and looped audio.
If you have an iPad, try Korg Gadget it can export ableton live sets also with midi.
I suppose it depends what OP wants. The Polyend boxes can print stems to the sd card. Of the ones I’ve tried it’s easiest to get from finishing an idea to dragging stuff into the DAW. And the new ones can multitrack into Ableton, though (maybe my settings were wrong) when I tried this it just crashed Ableton. Personally I just take them off the sd card anyway as it prints the reverb and delay as stems and keeps the main tracks dry. Not sure about MPC or the others, but I hear Roland does multitrack audio over USB.
Having had the same annoyance myself with Circuit I think the key thing is an easy way to process the audio tracks one way or another. There’s a whole category of grooveboxes that have no export at all, where as you say you would have to run each track out individually.
I chose an Akai Force when I had a similar aim - i.e make sketches / more developed tracks on hardware away from the computer then move over to the computer to finish off.
The Force can export as an Ableton project file, but I’ve not actually ended up trying that yet. AFAIK it’s fairly limited i.e obvs won’t have same fx or vst instruments available, but will happily get your midi data and audio clips / stems over as an Ableton project file.
I changed my workflow so that I actually multitrack audio and midi into Ableton or Cubase from my hardware rather than using the ‘export project as Ableton file’ method.
Force can have a class compliant audio interface connected over USB so you can set it up for multitrack audio IO.
Force is great in many ways - good value for a lot of power and features. Great inbuilt sounds and fx (with lots of great expansions and auto sampling of your current hardware or vsts making it possible to work with sounds from an Ableton synth away from Ableton for example). GREAT perform me control options - lush feel pots, pad fx, crossfader, macros etc etc.
It is a bit big though. MPC Live might be a better choice if size is a really important factor (and it runs on batteries with a decent speaker too!), but the Force has the clip workflow that I like and that I think you are looking for?
I know the OP isn’t looking for anything like an iPad app, but for anyone else looking for an Ableton-compatible groovebox, Ableton Note is of course very well integrated, and uses Ableton’s actual synth engines. You can also sample directly into its simpler or into drum pads.
And Drambo is incredible, and has built-in stem recording, so transfer is dead easy.
Another great option is Logic Pro for iPad. Logic has great instruments and FX, and if you also own it on the desktop, export to Ableton is trivial.