Short answer: Yes, use the 240p test suite and manually tune the sample rate for each console then save profiles. The linearity test is useful; also the test suite’s vertical black/white stripes screen (try to make all the stripes equal width).
The reason a sample rate of 858 was giving you a 3:2 aspect instead of 4:3 is because the OSSC’s line-doubled-240p output is digital 480p = 720×480 pixels, which when fully filled is 3:2 aspect. By changing the sample rate to 800 you’re putting black bars to either side of the 640×480 image, still inside the 720×480 frame. Different digital displays will display the 720×480 image differently, some may squash it to 4:3, crop it to 4:3, others may display it fully at its actual 3:2 ratio (sounds like what was happening with yours).
If you’re wondering what the numbers 858 and 800 have to do with the line widths of 720 and 640: The total “pixels” in each line of an analog signal equals the active pixels (e.g., normally something between 320 and 256 per line for a console intended to be viewed at 4:3) plus the back & front porch pixels, plus the sync pulse pixels. Together these normally add to around 400, which when doubled is the 800 sample rate at which the OSSC is set.
Changing the sample rate does seem to alter the sync parameters of the output 480p signal; and as you already noted with your display’s lost signal for odd-number sample rates, different displays will vary with their compatibility of said signal. So while you’re best bet is indeed to use the 240p test suite to tune each console’s width to your liking, there’s no guarantee your display will be compatible with the optimal width.
There’s an “optimal timings” chart on the OSSC wiki that is very easy to miss:
On that page you’ll find timings for N64, Saturn, PS1, DC, and a couple arcade boards. Personally I have not been able to successfully get the N64 and Saturn sample rates of 387 and 427 to work on my setup, though I’ve only tested on one TV so far.