This topic -adjusting valves on a 2F- has already been covered in great detail to the minutiae in several other threads. The reason the information can't be found is because the search function on this site continues to be virtually worthless. So we continue posting the same stuff over and over because nobody can find anything.
As for the Toyota recommended valve adjusting procedure for the 2F, the engine FSM is missing a word or two in the description.
The Toyota procedure recommends CHECKING the valve clearance while the engine is hot & idling. After a few valves are checked, (while the engine is running) the engine is then TURNED OFF to adjust them.
Then the engine is turned back on again to check what you did and do a few more that way: check while idling/adjust while off.
That's how I adjusted my valves for thirty years and never had a problem. I tried the method described above by the OP (engine always off) and my god what a pain that method is. I tried it once and never did it again - too much of a hassle with too much uncertainty for me compared to the engine running method.
The always off technique is useful when adjusting the valves for the first time after the head is rebuilt, but other than that (in my opinion) that method is for the birds.
Granted some guys have tried both ways and prefer the always off method, so it's best to try it both ways and make up your own mind. The end result is the same.
The one drawback of checking (not adjusting) valve lash with the engine running is it hammers the feeler gauge a bit, but I used the same feelers for over a decade & although they got a little bent (and I'd bend them a little to get a good angle anyway) their thickness never changed, so it didn't matter.
The benefit of checking valve lash while the engine is idling (then momentarily turning off the engine to adjust, then turn it back on) is that all the valve train stays uniformly hot and you can hear and feel how the valve clearances are.
You can hear if one is too loose (clicking louder) and you can feel if one is too tight because you won't be able to slide a feeler in the gap.
You can feel & hear the engine (while its idling) after all the adjustments have been made to verify everything is spot on. There's no guessing.