I already wrote a reply to this video elsewhere, if you'll forgive me the venial sin of cutting and pasting it here.
I've been geeking out over this sort of thing since 1993, when I bought the Star Trek: The Next Generation technical manual, an extension of the "Writer's Bible" created for the writers of that show to ensure they stayed consistent. They did indeed re-vamp the warp standard for TNG in 1987, creating a theoretical maximum of Warp 10. The explanation given was that power requirements increased exponentially as you went up the scale, making the achievement of Warp 10 something that would require all the energy in the known universe to accomplish. As noted in the video you linked, writers occasionally got around this by implementing a phenomenon other than standard Federation warp technology.
In my own books, I made a critical error very early on in that I wanted the ship to visit the Large Magellanic Cloud, some 169k light-years distant. For plotting purposes, I wanted the trip to take six weeks, so after crunching the numbers that means the Reckless Faith would have to go 1.5 million c, a velocity of which I'm sure the Federation would have been quite envious. What I hadn't considered is that when I moved the action back to our own galaxy proper, this meant the Faith could essentially be anywhere in our quadrant in mere days, which makes it difficult to build tension in certain situations. As a blatant act of retroactive continuity, I took the liberty of manufacturing another limiting factor: time dilation. Though one much smarter than I could use math called the Lorenz transformations to actually calculate time dilation at velocities like that, I used author's fiat and fudged it. However, I stayed true to the fact that the faster one goes, the more time passes elsewhere, and the limiting factor was that they had to strike a balance between speed and time. Go too fast, and the problem might be long over by the time they get there. From book 3 onward, they only go their top speed if it's a short distance, or time is not of the essence.
Even still, the Reckless Faith is supposed to be the exception. Most ships are limited to 800 - 900 c, but for plotting convenience I've also allowed 2500 and 22,500 c. As long as those numbers make sense within the context of the story and the ships to which they apply, I think I can get away with it. On the plus side, weird alien technologies notwithstanding, I've allowed myself a much larger playground within the galaxy than the Enterprise ever could hope to access.
