I don't think that topic is what the OP is asking about. That was in regards to the draw pile running out.
To answer OP's question. The second action would not be taken and would "fizzle" with no effect because it cannot be completed. I find this comes up quite a bit and it can be annoying.
One thing I don't think was mentioned in the rules (please correct me if I am wrong) but IS in the video is to flip ONE card into the discard pile for EACH player before the game. So if there are 4 players you'd flip 4 cards into the discard pile, 3 players 3, and 2 players 2. This helps a little bit in the beginning but the discard pile can still get depleted. The beginning of turn flip one mechanism also helps.