My two cents.
The Brummel splice shown in the video is fine for kite lines. It's almost impossible to make them to a specific length, but easy enough to make them consistent so you can make all 4 lines match by measuring them the same.
If you want to make one line match another then the normal sewn splice is better because you can easily adjust one line to match another line or a measurement then pin it and sew it. You can still lose a few mm because the bury takes up some length in the line, but it's not so much over the length of a kite line. All the instructional videos show the bury being tapered. It's easy to do so why not? I have no idea if it actually affects the strength. As I said, it's important to make the buries the same length if you want consistent line lengths.
To measure the lines accurately you should anchor all four to a solid point then equally tension each line with 5-10kg in weight. Ideally that would be separate weights for each line but near enough will be good enough. When you tension the lines they will all stretch to a point until the weave locks then they won't stretch much more. If you're observant you can do it by hand pressure.
You don't need special tools. A bent guitar string or thinnish bit of wire is good enough for most line thicknesses. For super thin line use a darning needle from a sewing shop. They're about $2. The thinnest line I have successfully spliced (sewn splice) is 0.8mm.
When I used to make bridle lines I made a measuring jig out of a length of timber and had nails to mark the lengths of each part of the splice.
All this was covered here
www.seabreeze.com.au/forums/Kitesurfing/General/How-to-shorten-kite-lines-?page=1