By Kriyas Yam Suf it says that the first one to Jump in was Naḥchshon ben Amminadav. My question is: why didn't Moshe Rabbenu jump in first?
I believe the question is backwards. The question should be, why does the midrash suggest that Nachshon, was the first to enter the Yam Suf?
It seems, that the midrash about him entering the Yam Suf, was a response to the fact that Nachshon gives the offering to the Mishkan first, while earlier, he is listed 5th. This is explained by saying that Moshe could not decide who should go first, but all the other princes and leaders pointed to Nachshon, saying that since he went first into the ocean sanctifying Gd's name, then here too, he should go first and sanctify Gd's Mishkan.
As to why the sea did not split until somebody walked into it, perhaps it was a counterpoint to the fact that Gd just fought a war for the Jewish people, and soon the Jewish people would have to fight wars themselves. A sign of trust in Hashem to split the ocean was needed by the Jewish people to prove that we could "open the eye of the needle", before Hashem would "create an opening as wide as two camels." Since Moshe had direct knowledge of Hashem's abilities and miracles, he himself could not fulfill that function, and some other leader of the Jewish people would have to do that.