Some movies in Keynote presentation won't start automatically

How is the projector connected?

and go into presentation mode

Do you mean Play or Presenter Display?

Two workarounds to try:

1 Use play in a Window and use the window full screen

2 Use another Mac Account as shown below, does the problem still happen in the different account?

Set up users, guests and groups on Mac – Apple Support (UK)

I have a very similar issue, although it doesn't seem to matter whether my computer (MacBook Air, 16GB running OS Monterey 12.3.1) is connected to a projection system; it happens when I'm playing my Keynote presentations on the computer itself. I first noticed around the time of the update to Monterey 12.3.1 happened in April. I gave a presentation on April 24, and some of the slides with videos would not play at first. (The slides with static images only always do play.)

I'm working on a new presentation now, and the slides with videos don't play the video about 10% of the time. I have to backtrack to the previous slides and then try again. Most of the time that jump starts the "stuck" videos, but it's a distraction and throws off the timing of my presentations. I suspect that there is some kind of compatibility conflict between the latest version of Keynote and the latest version of Monterey OS. I hope that a fix will come soon, because it's problematic for those of us who rely on Keynote for our presentations.