Just wanted to close the loop on this and provide a solution for anyone else who runs into this problem. The plugin Discussion Board Pro does, indeed, seem to be the source of the conflict, but its developer cannot figure out why either. Its code should not be impacting any pages other than those where the Discussion Boards live – and that is not where the videos in question live.
However, we did figure out that adding this CSS to the global CSS fixed the problem:
.mejs-container {
min-height: 600px;
background: white;
}
Obviously, the height can be set to whatever works, but not having a minimum height is what made the video not appear. In addition, with the Discussion Board Pro plugin installed, the background of the player was a dark gray. I changed that by setting it to white, but it could be set to whatever color you prefer.
Thanks again for your help here!