As I understand it, a black-and-white TV tube with no color burst signal and the RF stage skipped, while it could not quite reach 80 columns, could handle 64 columns well enough,
It can do 80 as well, but that's already borderline.
which raises the question of why Apple and Commodore didn't try for that, or at least for something a bit beyond 40. 64x25 = 1600 bytes, not prohibitive even by the standards of the time.
As Stephen Kitt already mentioned, 1 KiB is a lot of memory, especially considering that the early machines like PET and Apple II were meant to work with as little as 4 KiB total RAM. The 'mere' 600 bytes needed for 64x25 would have been a rather deep cut.
Where Commodore and Apple stayed with a 'traditional 24/25 line display, Tandy did go a different way by making the screen 'binary' compatible with a 64x16 resolution - thus staying within the 1 KiB needed for display memory but showing longer lines.
I get the impression memory bandwidth was a significant limitation, such that when the Apple II later added an 80-column option, the engineering of such involved the careful juxtaposition of two memory banks to provide data at the required rate.
Not in any way. Simple math tells that a 1 MHz machine with video a video access window on every clock can deliver 1 MB/s. A 40x24 screen at 60 Hz needs less than 60 kB/s, that's less 6%. There's much room for a larger display, even when including other constraints like frame structure.
The problem with the Apple II was that its timing is built around the video timing. The CPU doesn't get a monotone clock rate, but a varying one to make it fit. While this was a quite clever design, it's somewhat inflexible when data needs to be accessed at double the rate. Ofc, they could have went along and redesigned the whole timing - or doubled the machines speed (which would have been nice). The least intrusive way to double accessed data within the existing timing was adding just another KiB and make video access 16 bits wide.
Was memory bandwidth the reason for Apple and Commodore sticking with 40 columns?
No. On the Apple side it's rather clear that WOZ intended the use of a TV set with (or without) intermediate RF. Thus 64 would be out of scope and 40 seams a nice thing.
For the PET, I can only guess that they tried to stay on the safe side (*1) - and, maybe more important wanted to keep 24 lines. With 24 lines 42 would have been possible - but that's really odd, isn't it?
We often forget here the other early machine, the Atari 400/800. Here, like with the Apple, a standard TV set was intended as main display, so 40x24 is a logical choice.
If so, how did Tandy achieve 64 columns? Is it purely a coincidence that the 64-column machine is the only one of the trinity to use the Z80?
As soon as one is willing to diverge from the 24/25 lines paradigm, 64x16 is a quite reasonable choice. Especially when it comes with an 'internal' screen
(I'm assuming the Vic-20 later supplying only 22 columns, is not so much due to technical limitations as to the use of a video chip designed more for games than general purpose computing.)
I think that's a good assumption.
Much like the TI 99/4 was designed around the use of a TV set. Here the standard mode was only 32x24 (a two color 40x24 was also available).
The C64 later on benefited from the 40x24 decision for the PET.
*1 - I would like to think that they did keep an eye on the use of a TV during design - and before the decision was made to produce an all-in-one unit.