My 2001 Sequoia SR5 has the similar issue, any update on your car?
Finally, after four trips to the dealer, replacing the brake pedal sensor we may have found the answer....
Despite the alarm and the lights coming on another issue came up - I couldnt shift the car into Park (which also meant i couldnt turn the car off and lock it).
It took numerous pedal pumps, key jiggles, waiting and shifting around to get it into park. The dealer thought it was the pedal sensor and replaced it - problem persisted, sporadically.
Every now and then without the key in the ignition and the car off, I noticed the gauges starting to flicker, needles dancing and the cruise control light would come on.
I was at the trunk and heard the car turn on and a beeping sound (like when you first turn the key to ON).
Another time I parked the car, pulled the key out of the ignition and was checking my phone when i saw the gauges turn on and the beeping came back (and the DRLs came on). I took the key and gently poked at the igntion cylinder and instantly everything died.
Im convinced that the issue has to deal with the ignition cylinder. I recalled one instance when I couldnt start the car after pumping at a gas station, after disconnecting the battery and playing around it finally worked.
after researching the issue (which seems seldom) Ive learned theres a sensor behind the cylinder that detects that the key is inserted, that could be bad (preventing me from putting the gear selector into P). alternatively the contacts in the cylinder could be corroded or loose causing the issue.
Will post more info once i tear into it.