How do you guys get from Round 1 to Round 2 without showing the Digimon animation where "You Win" appears on the screen?

I try to press start or the buttons and it doesn't work. Do I have to unlock more than just Reapermon in order to do this?

I'm having trouble understanding what these main quests are. It would be clearer if you said the name of the last mission to be done.

I can use Arcade stick as controller in this speedrun?