Car is completely dead until battery disconnect/reconnect - then fine

2003 RX300, with 110,000 miles or so. We've owned for a few years, since
90k or so. No issues, other than battery dying if we let it sit for several days
without driving it. We bought a battery quick disconnect that mounts right on
the battery terminal to fight the passive drain issue, and that has worked fine
for last 2-3 years or so.

Recently we saw the VSC/Check Engine light come on intermittently. Since
we disconnect the battery and leave it that way until next time we drive car,
that seems to have lessened.

But now a new issue has emerged: if we leave the car sit overnight without
disconnecting the battery, the next day the entire electrical system is down.
No remote locks, no interior lights after manually unlocking/opening the door,
no dash lights with the key in the ignition, and no starting.

Oddly, this is "fixed" by simply opening the hood ('bonnet'), disconnecting the
battery with the quick disconnect, and then reconnecting it. Lights come on,
ignition key warning bell sounds, car starts right up.

So why would cycling the battery connection 'fix' this issue? What would
someone do if they didn't have the quick disconnect? Who'd a thunk that
would work? It appears like a completely dead battery, but that's actually not
the case.

I'm reluctant to take it to a dealer - sounds like lots of bad experiences getting
charged to replace all sorts of stuff that may or may not address the issue.

Thanks for any ideas.