Found out that Baird used synchronous motors after it's been proposed that the US should have a phase synchronized 60Hz grid.
At first, the electricity companies didn't want to have an additional burden for zero benefit, but it's been adopted very soon, as it proved beneficial, because they could use the grid to compensate for failures and power lacks.

I decided to go with a synchronous version for the first try. I was able to source two synchronous motors (however untested). The one on the right looks period correct (early 30s ?), the left one is a backup.
Let's see if they work. It's the best substitute I can get, and it's what they were able to buy off the shelf back in the mid 20s.

They will lock onto the 60Hz which will make things MUCH easier. However, you can't guarantee the phase at start. Depending on the number of poles (4,6,8), it might lock in a different position.

I guessed this could be compensated for by either rotating the whole motor assembly (has been done), or by briefly interrupting the power supply for a quick moment.
And yes, looks like Baird did that as well. There was a push button that the user had to tap a few times, until the motor locked to the correct phase.
I found a picture where he uses a synchronous motor in his late 1920s setup that looks pretty much identical to the one on the left.

Fingers crossed that those motors are OK, as finding an old synchronous motor of that type is not easy.