In that guy's system, cutting the card actually made some sense, since that was a super cheap $40 graphics card even when it was new, and he was installing it in a dual-socket server motherboard that undoubtedly cost far more. I doubt that low-end card would have even missed the bandwidth, and he was mainly just installing it to enable higher resolution display output anyway.
As far as Linus cutting out the end of the slot in the other video, a dremel might be an option, but it would probably be better to use something like a smaller cutting bit rather than a big cutting wheel that will likely to mess up the pins and fling plastic dust everywhere. : P