I'm sure the author of that video is competent in other ways, but not very competent in soldering coax connectors on 'foam' coax. The only 'trick' is applying enough heat for the shortest amount of time.
There are several 'trick' that can be used. Probably the easiest is to pre-tin that reducer, makes things a lot easier and takes less time to solder to it correctly. Another 'trick' is to cool that connector after heating it.
As far as not soldering that center conductor... Why not? Same things, proper amount of heat for the shortest time. Then cool it.
As for the double shielded coax. Leaving that foil exposed so that it coul make contact with anything is not exactly the smartest thing you can do. Trim it even with the 'folded-back' braided shield.
One 'trick' with doing the soldering on the center conductor is make that inner insulation 'work' for you. Assemble the connector and before soldering the center conductor very, very slightly pull it out of the connector just a smidgen. That makes the inner insulation of the coax sort of pull up into the center pin (not really, but it does eliminate any space between the coax's inner insulation and the center insulator of the PL-259. Keep that center conductor pulled out till you've cooled that solder joint, then trim to length.
Preparing surfaces that will be soldered is a very good idea. Clean, abrade a bit, and use flux.
Watch someone who'z done this sort of thingy and you'll see most of those 'tricks'.
And lastly... It make a big difference where you do all that soldering on of connectors. If you are out of the weather (which means wind), a large'ish soldering iron/gun work okay. If you are out in the weather (wind), you'd be better off using a torch. 'Course, that means you have to know how to use that torch to solder to start with.
All it takes is practice. If you just happen to have one of the old-time Amphenol PL-259's with the 'bakelite' center insulation, you might be surprized at how often you can practice with them without melting anything. I love'em! I've re-used one or two of them longer than some people here have been alive.
And no, I'm not an expert.
- 'Doc