I disagree that your problem has anything to do with the built-in graphics cards falling short. In my opinion, you are trying to move more data to the six screens than can be transferred from the number of sources you have, in the time you want.
Try this experiment:
Set each of the displays to mirrored (the same data on each screen). Now play ONE movie onto all six screens. If you get drop-outs, the display card is defective (or, as you assert, inadequate.)
I believe your other experiment (with six different data streams to six different displays) is I/O bound. If you get drop-outs you need more drives in more different enclosures to provide enough concurrent data for the displays.
Another experiment to try is to run two displays. Then try three. Then four, and so on. If you study where it fails and analyze how much data is needed to support each stream, I believe you will find that the amount of data required is more than can be produced by one or two ordinary drive or SSDs.
A single ordinary rotating drive can produce a single burst at about 125 MBytes/sec, not sustainable. To supply moving pictures to six displays takes far more data than that.
A third experiment is to reduce the resolution on each display to the lowest it can achieve, and play those smaller windows. Perhaps it can play six half-sized movies.