The Browning Hi Power is a fine pistol, and for its own time, was quite a masterpiece, that it was one of the earliest "Wondernines," that combined a nice capacity with excellent reliability and function.
Even though it was an all-steel weapon, it still concealed quite well, having a fairly flat profile, and some really nice ergonomics (after a few refinements throughout the years). I remember the joy of shooting one of the BHP's with the Portuguese stamps on it. That weapon had a fine, crisp single action trigger, and felt wonderfully balanced.
In today's applications, though, it only makes sense that the British military would switch to the Glock 17. You get 4 more shots, while shaving off 13 ounces of weight, while also using a much simpler (yet effective and reliable) design. There's no doubt, that the cost of the weapons in the long term, are going to be much lower than what the old BHP would cost. It also simplifies the user requirements, since the double action only equivalent functionality doesn't make the operator think about whether or not the safety is on, whether or not the weapon is cocked and locked (the video stated that they couldn't carry it that way?), etc. There's only one state, that it's ready to fire with a simple trigger pull.
Overall, the video is fairly accurate, except in regards to the thing about accuracy. In terms of accuracy of the weapons, I do dispute their claim that the Glock 17 is more accurate. I would actually give the Browning Hi Power a very, very slight edge in accuracy, due to the excellent single action trigger pull.
Of course, that ever so slight edge means nothing when it comes to combat accuracy, and that in all honesty, either weapon is going to have a greater capacity for accuracy than the average individual user.