For future readers, I have replaced the tilt motor with a unit from Toyota. The updated part number was 89231-60022. There were only minor changes to the design.
The installation was not without challenges. First of all, mucking around with the old motor had resulted in the fault codes being set, so that the "Tilt and Telescopic ECU" would not try to run the motor. This meant that the new motor didn't run when I first plugged it in, which was a little disconcerting. However disconnecting the batteries for half an hour reset the ECU and fixed that problem.
I also found that the motor had to be screwed in to the threaded adjusting nut exactly the correct number of turns in order to have the steering wheel tilt the full range. The motor only does three revolutions and then stops, in either direction. By correctly fitting the motor it stops before the circlips on the threaded screw hit the threaded adjusting nut. Correct fitment means that the motor doesn't try to continue rotating after it has hit the hard stop of the circlips, which could result in triggering a fault code, or damaging the motor.
I couldn't remove the threaded adjusting nut from the steering column to clean it out, and since I wanted remove most of the old grease including in the adjusting nut, I screwed the old motor into the nut and cleaned away the grease it pushed out. Since I didn't care about the old motor, I screwed it in as far as it would go, and about half a turn more, which of course rotated the spindle in the motor. The spindle of both the old and new motors can't be turned by hand.
When I had unscrewed the old motor I plugged it in again and surprise surprise it worked. :o Not as well as the new one, but it did rotate three full turns in each direction, the same as the new one.
So, either the old motor always worked, but the fault codes were set so that the ECU didn't try to run it. I don't think this was the case though, because I had had the batteries disconnected for extended periods previously, and the tilt motor never started working. The other possibility is that by pulling it apart and reassembling, then rotating it half a turn, the motor was reset back into a functioning starting point. That is, previously it had been rotated out of its range of operation somehow, perhaps because of an overload cause by lack of lubrication, and rotating it back into its range of operation enable it to work again.
I did have Lexus look at the motor once because it was a bit noisy, so maybe they pulled it out, incorrectly reinstalled it, leading to an overload, or they just rotated the motor out of its operating range. They looked at it a second time when it stopped working, and declared that the motor had failed. Now I know better.
Anyway, I installed the new motor, since I had it, and the old one can be a spare, or get recycled some time. The lack of solutions offered in this thread would imply that this is a rare issue. Maybe other Lexus dealers know how to grease a thread, and correctly use the diagnosis tools available to them. The tilt motor seems to be pretty robust if understood and treated right.