But, hypothetically, within those limitations, some sort of USB host would still do those things?
Not that I am aware of. My EQ2 is controlled directly via midi. In my case a Boss ES-5. All the other Source Audio devices that I use are controlled by my ES-5 via a Source Audio Neuro Hub. My UltraWave and C4 receive midi commands via the Neuro Hub, but I don't bother with scenes with these two devices. I use the Hub to relay midi from my ES-5 to the pedal's internal patches (midi PC change). If I want to call up a patch on my C4 or Ultrawave, I send a Midi PC change command to the pedal and the hub passes this on really well and it's as if the two pedals were directly controlled from my ES-5 via midi. The same system also works for Expression via CC midi.
My Lunar and Aftershock are both older One Series models and this approach doesn't fully work the same.
Both my Lunar and Aftershock are connected and controlled by the Neuro Hub. However, these two devices do not have Midi PC capabilities. So I can't send patch change over midi to either device. It's a limitation of the device and it's firmware. It's by design and can't be easily retro fixed. These two devices are mature products and can't be re-coded without a large risk of breaking all of the current settings / patches that are on the device and on the community cloud.
You can select the model on both devices, just not the 6 pre-sets / patch. So I can use midi to change / choose which modulation model on my Lunar and which distortion model is used on the currently selected patch. But none of the other settings are touched. Not all of these settings can be controlled via Midi, but most can. However, sending that amount of Midi commands is a slow and cumbersome way of changing a patch on either device.
The way that I use extended patches on my Aftershock and Lunar (beyond the 6 internal) is to utilise the Scene options on the hub. The hub has 128 patch memory that can be spread across all SA pedals attached to it, up to a max of 6. Before I had my Lunar, I used the Hub Scenes purely to service my Aftershock. When I got my Lunar, I had to rethink how I was going to use my hub. I have 8 sounds that I regularly use on my Lunar. So I set up the 6 internal pre-sets with my top 6 most used. I then copied these over to the Hub (not and easy task) and reserved patches over No#40 to be for the Lunar. My 1-6 from my Lunar are loaded as 41-86 on my hub. 47 and 48 are my other two patches that I like to use. This means that I can access my top 6 from either the device or the hub. My ES-5 calls up the Scene via the Hub and I have the other patches are available via the device 3 ways switch in Green or red mode.
My Aftershock is configured in a similar way with my hub. Scene (Patch) 1-6 on the hub is identical to my 6 internal patches on my Aftershock. I currently have 37 Aftershock Scenes loaded on my hub. Most of these Scenes are dedicated Aftershock Scenes. Interestingly, If I call up say Scene 12 and then Scene 47 from my ES-5, the two scenes don't over write each other. So both scenes effectively become live (calling up a Scene that just uses the Aftershock, doesn't over right the Luna settings and vice versa).
4 of these Scenes are very specific dirt used with an OC-2 patch on my C4. These 4 scenes also have Lunar / Modulation patches in the same Scene. So these 4 scenes control settings and patches for both pedals at the same time. I have 2 Scenes that have Aftershock, Lunar, Ultrawave and C4 patches loaded. So I can call up one Neuro hub scene and it calls up 4 patches on 4 different devices.
This system is really flexible but requires a bit of thought and configuration to get working correctly. I can call up any of the internal 128 pre-set patches stored in my C4, Eq2 and Ultrawave via midi. Each device also has it's own 6 internal pre-sets I can easily change manually. My EQ2 has 8 internal patches. My Aftershock has 37 patches loaded as Scenes in my Neuro hub and can be changed via midi. My Lunar as 8 patches loaded as Scenes that can also be changed via midi, I've also got four Scenes that load both Aftershock and Lunar patches.
When I had my Neuro Hub only controlling my Aftershock, life was a bit simpler. I used the hub as a way of added 128 patches to my Aftershock. When I integrated my Lunar into my board I did consider getting a 2nd Hub (on it's own midi channel) to add 128 patches on to the Lunar. This is a very viable option and it works well. The hubs are fairly cheap, small and easy to power. However, I soon realised that my Modulation patch needs were actually quite small (although essential). I find the max 6 internal patches a bit limiting on the Lunar (strangely less of an issue with the Aftershock...go figure). But I find that I only need less than 10 different modulation patches. I think that modulation patches are more varied in sound and application but there's less of them. Where as Dirt (on bass) is less varied in it's sounds but more plentiful in type. I can get by with less dirt patches than I can with modulation. I have a UniVibe, 2 phasers, a Leslie cab, 3 Chorus and a Flanger set as presets. I just don't use any more than that (but I can if I want to). A univibe sound wont cut it on a patch / sound that needs a gentle Quad Chorus. Where as with Dirt, I have way more options in terms of dirt models...but when applying dirt to stacked effects; one light overdrive is very similar to many others if I tweek the knobs a bit.
So to answer your question...no I don't think you can add a USB midi controller directly to your Aftershock and change your Aftershock patches. You can get it to change it's dirt model via midi (but retain all the same patch settings). It's easier to plumb in a Neuro Hub and use it's Scene array to add 128 patches to the Aftershock's work flow. But you'll need a midi controller and a Hub. You could then use your hub to control other SA pedals...or have one hub per SA device. It's up to you how complicated / simple you want to make it, which is one of the great advantages of the Source Audio Eco-System: it can be super easy...or very very deep dive if you want to.