The chain is the big one where you're going to notice a clear difference if it's not lubricated correctly. Even then, how much rain it takes to actually wash out the lube depends entirely on what you're using and how much is on there. Over-lubricating your chain is one of the most common mistakes in bike maintenance. Particularly if you're using a wet lube, you want to know what a dry chain looks and feels like and re-lube then, but not much before. Dry lubes are less sensitive to this - because their liquid part is mostly (depending on the exact lube) a volatile solvent that just evaporates, over-lubricating or haphazard technique won't result in the chain being a sticky mess, or at least not as readily.
For derailer and caliper brake pivots, the differences are mostly incremental. Lubricating them will make things a little bit smoother and is a nice way of keeping things working their best, but usually won't make the critical difference in whether they work period. I do it by putting a drop of light viscosity wet lube like Triflow or Dumonde Lite at each pivot, and I use that pretty much regardless of riding conditions. I find that thick oils don't wick in there very well and dry lubes don't stand up to rain very long, although they're fine in a pinch or for dry riding. I also put a drop at the gap of each RD pulley to help prevent squeaks. That's a spot that can be over-lubricated, but a drop or two is good in general. Some pulleys are much more squeak prone than others. I generally don't lubricate mechanical disc caliper armatures because the actual pivoting surfaces are pretty far down in there, and often it's a cartridge bearing anyway. Lubes that are designed to wick and migrate freely are also a good thing to avoid getting anywhere near a disc brake if you can help it.
The big problem with getting aerosols involved anywhere is it's very easy to get lube where you don't want it, like rims, rotors, and brake pads. In the case of blasting entire brakes and derailers to lube the pivots, you're leaving some kind of oil coating on everything else, which will gather dust if it's dry and get washed off (potentially onto your braking surfaces in the case of a rim brake) if it rains. That said, a lot of people do it to derailers with no real ill effect, although I'd still recommend careful technique, and using a rag to shield everything else. As for the video you link to, you can actually see him hit his rims with the WD40 when lubing the RD, and lubing brakes that way is really asking for trouble. WD40 itself is a reasonable lube for pivots as the solvent part evaporates and leaves an oil coating behind.
The other piece of this is cable lubrication, which can be one of the more pedantically controversial topics of internet bike mechanical discussions, and also an area where what works best is most bike- and conditions-dependent. Lubricating cables always introduces some propensity for them to attract grit, which has lead some to say just don't do it, especially since modern housings have slick plastic liners. While over-lubricating or using the wrong lube can certainly cause problems, I find that 9+ speed bikes with normal non-continous housings need it on their RD gear cables at least to work right consistently, particularly in the rear RD housing loop. I live somewhere where rain is ever-present, and my preferred housing lubes for all bikes are the dryest and least sticky of dry lubes, like White Lightning Clean Ride (formerly White Lightning) or Finish Line Dry (formerly FL Tef Plus). I find that these don't really ever cause problems to speak of with drawing in grit. Using a very tiny (like half a pea) dab of very light, slick grease like Slick Honey on gloved fingers to coat the cable under the RD housing loop also works well in wet environments, again because it's a bit of a potential trouble spot and having something a bit more tenacious works well. As it relates to the question, I'm not saying do this all the time, but riding in heavy rain will make you need to check cable lubrication more often.
