Given the history, forum posting history, for the 2001 RX300 if mine had that kind of mileage I'd be overhauling the transaxle.
Have you checked the ATF fill level or condition..??
ATF should be mostly clear/transparent and light pink in color. ATF looks great and it is full. No traces of metallic fines or discoloration.
With the ignition switched on but no engine start make sure all the instrument panel diagnostic lights/icons illuminate. All warnings lights illuminate.
Is there anything to the recommendation of cleaning out the resistors down stream of the air filter?