I've been using Gambatte to speedrun my games, but for some reason the A + B + START + SELECT does not work. Does CTRL + R count as a soft reset too? Why wouldn't the usual combination work?

You generally need to hold down the keys for a few seconds before the soft reset takes place. In some cases it will take longer as the game can not be soft reset during certain circumstances, such as when a Pokémon is currently fainting.

In the NSC WR run the video shows the game being reset immediately during the brock skip. is the runner holding down a+b+start+select during the save and it resets immediately after? I know for a fact that the NSC runs seem to have no +5s on emulators based on submitted times

My knowledge of NSC is very limited. But from looking at the leaderboards, we see that the current WR (17:06 by Pokeguy84) was done on a Gameboy player, so I would assume he used a+b+start+select. You will most likely be better off asking him directly however. Also, a fair share of the top runners of NSC are very active in the community and should be quite easy to get ahold of through Twitch or Twitter.