The video was NOT the Gulf Stream.
I ran charters to the Bimini and the Bahamas on 67' sailboats, that were more or less motorsailors.
The stream can be a scream whenever there is any sort of northerly -- the winds and current in opposition. That is when the seas are what we used to call "Square Waves" of 15'-20' where as you came off a wave the boat would just drop straight down as if off a cliff.
At 8 knots I would steer 15-degrees farther south and wind up right at Bimini every time, dead reckoning -- all we had then was Loran, which did not work in that area.
Best launching spot would be somewhere like Key Largo, where you would have the stream pushing you all the way.

And Bimini is a blast.., good restaurants and bars. End of the Earth Sand Bar was pretty funny. Nice people as well.
Have fun,
Rick