Highlander Random Non-Starter

My 2003 Highlander has developed a random non-starting problem last August; each time, the engine cranks but will not kick over. When it does not start, five times now at between one to three-week intervals by end of September, the car was towed to a different mechanic, including three times now to a Toyota Authorized Repair Shop (by now two different Toyota shops). Nobody has succeeded in diagnosing the problem correctly so far because while in the shop, the car starts every time. Of course, the car is never kept in the shop for the same interval as I drive until it fails to start, nor is it started as frequently as I need to during my customery use of the car. Suggestions made as a best guess cause, which I carried out but which did not fix the problem, are: change the key; change battery on keyless entry device; change the car’s battery. An additional problem is that the dashboard light that usually goes on when the keyless device is used to lock the car, now does not go on; however, mechanics don’t think that this faulty circuit has anything to do with the random non-starts.

Has anyone experience with a problem like this, and if so, was a root cause found?