Yes, I've seen the video. In the video, they mentioned the need for a low temperature melting point and then they notably left out petroleum jelly that has the same melting point as the original Singer lube (the UK formula, I suspect because the US one seems to melt higher than 115F)
They and the other person selling a reformulated lube say not to use PJ but others who don't sell their own product - as well as smarter people than me - say PJ is fine.
Without testing the reproduction products personally, I can't give an opinion on them and I'm not purchasing them because the price landed in Canada will be between $35 and $40 per tube. It is worth noting though that many people have been using PJ for years as well as White motors with grease tubes calling for PJ.
The electrical motor shop told me today that unless the wicks are really different, there's fundamentally no difference between a Singer motor and a White motor as far as their requirements.