Well, now that I've taken delivery of my 2021 Renegade Trailhawk and searched my USB-based 2020 Service Manual, I can shed some light onto the auto stop/start feature -- at least the U.S. version. Particularly the question about the "second tiny battery."
When driving the vehicle home from the dealership, I noticed a warning on the dashboard -- something to the effect that the auto stop/start system was disabled because its battery hadn't charged up enough yet, or something to that effect. By the time I got home, there was a notice that the system was now working.
Nothing about this in the Owner's Manual. So I just searched through the Service Manual, and found some details on how the system works. Since the weird copy protection prevents me from cutting-and-pasting or even doing a print-screen on anything from the manual, here's an old-school literal screen shot (* grumble grumble pain-in-the-ass *). Hope they don't get me for a copyright infringement (coughJeepCarescough
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So it looks like there is indeed an "Auxiliary Battery" for the ESS system. I haven't digested all of this yet, but it looks like the Auxiliary Battery kicks in when the engine is automatically stopped, to power all the vehicle's electrical systems so that the Main Battery isn't burdened and is ready to start the engine again. So if the Auxiliary Battery isn't working or isn't charged, the ESS is disabled. Pretty smart.
Also, according to the owner's Manual, there's a long list of conditions that will prevent the ESS from engaging -- high-power-draw stuff like the HVAC system not being at its set level, or seat heating on, etc etc etc -- all to make sure that power isn't drained when the engine is off. Again, pretty smart...