To add to why Time players are necessary, setting up paradoxes is necessary to creating a frog, and a Time player is often needed for this.

Paradoxes aren’t needed to create the genesis frog.  The time paradoxes in Karkat’s session had nothing to do with the making of the frog. The Beta Kids’ were very crunched on time since their reckoning was going to start early so they needed the time travel to finish the quest.