Wow, I tracked it down, and I have never taken mine apart like that, and I’ve had it for like 6 months almost. I didn’t even know you could disassemble the pump like that. Yeah, that dude says that part is important, so maybe you should contact the company and see if they can send you a little sponge piece. Sorry I can’t help a bit more.
Edit point: I just took mine apart, and mine doesn’t have a sponge in there either. just a little rubber piece with a squiggly/squishable part, attached to the square peg that sticks out the very end. So I’m thinking that video is of the original model, and the design has been updated to not need that sponge piece. So you’re good to go. One less small part to lose I guess.