There's plenty of time to shift and the shifts in the Rigoletto "solo" are not that difficult (unless your shifting is not accurate), so what is there to gain by crossing strings? So, yeah, Barry gives good advice.
I'm perplexed though at why he suggests doing the C-F-E-D-C-C at the end, once on the G string and once on the D string. I can't think of any context where that would be a wise choice. It's high risk with low reward.
All this being said, Barry's videos are great. They were one of the first of their kind and very innovative and well-produced for their day (well-produced even by today's standards). Back in 2013, most bassists were still too afraid to put themselves out there on video playing excerpts, much less even wanting to give lectures about them. To this day, as many major symphony musicians from Barry's day are retiring, they have gone their entire careers with most people not knowing what they really sound like on excerpts except for the students who take lessons with them. Different generation for sure, but Barry was thinking like a young guy.