For anyone following this topic, I experimented with several different analog pedals, either in front of the POD Go or in the FX loop, depending on the appropriate use case. These included the Xotic EP, SP, and Super Clean, and a Keeley Mini Comp. None of them made an appreciable difference in the output, although they're certainly great quality and do what they say on the box. I particularly liked the sound of the Keeley (and the EP was too dark for my tastes).
I did resolve an issue that was affecting the POD Go's overall sound, where some very nice commenters pointed out that I had the Master volume on my amps all the way up, which was essentially getting in the way of my tones. My main guitar has a broken volume pot/knob and is stuck at 10, so it was driving the amps too hard and making everything sound muddy.
I'm not against trying some other analog pedals or dropping one in for a specific tone (or to save DSP), but I also didn't find anything to suggest that having one improves the POD Go's sounds altogether just by being there.