You're right that the "doesn't end in white" problem has a lot to do with "extending to 7 spaces"! If you go back to the video at around 1:43, you can see that Professor Loh starts talking about this. He says that using "B" and "WB" blocks can get you a lot of good strings of 6 beads... but there's the slight problem that it doesn't count the ones that end in W!
And that's why we extend it to 7 spaces! Why does this fix the problem? Well, those 7 spaces will always end in black, right? So, the actual "things" we are considering are all those combinations of beads in the first 6 spaces! You can see that the 6th bead could definitely be white, so that solves the issue.
Does that make more sense?