I took a chance on a dead
ES-1 mkII "RHYTHM PRODUCTION SAMPLER" from eBay and no surprise it didn't work when connected with the correct power supply.
Carefully comparing the schematic for the ES-1 mkI (I note it's "Provisional Version, Revision: 1.00" and therefore likely a leaked pre-production copy) and checking voltages on the board I was only getting +0.79V where I was expecting to find +5V for the DIGITAL section. I didn't spot it at first, but eventually noticed that the
S8520 step-down switching regulator-controller is toast (
IC39 on the board):

Not as dramatic as the melted mess on
feeedback.loop's ER-1 mkII, but a very similar failure. They're protected from reverse polarity, but I don't think these devices can stand an over-voltage for too long. Spec sheet says 18V is the absolute maximum the
S-8520 can handle. It's not clear if
F3, the CPH6302 P-Channel MOS Silicon FET, is dead too, but neither of these things are more than a $/£/€ or so.
I note that the code on
IC39 is
BOJU, which makes in a
S-8520F50MC-BOJT2U which is the S-8520 Series F, 5.0V output, 300kHz Osciallation Frequency with No Overload Protection and "Lead-free (Sn 100%), halogen-free"!
For future reference for anyone else that finds themselves in a similar situation, here is a photo of the power supply area. Note that this is the
ES-1mkII KLM-2507 board. I believe it's functionally the same as the mkI, but the board itself has been revised because the component numbers don't make the mkI schematic:

And here is the PSU section of the schematic. Again note this is the
mkI schematic for board
KLM-2197, which I have overlaid with
mkII component numbers in
red (use at your own risk, this is my best guess tracing and inspecting the board):

Thanks again to
feedback.loop for his invaluable YouTube videos.