There is a bit of a problem arising from the fact that Reggie replaces S.T.E.V.E. as The Great Turkey. Since Reggie returned before he left, we are inside that first anomaly; in a few hours, S.T.E.V.E. will be launched, and so we are facing the end of this history (whatever happens from that). The two anomalies interact in complicated ways.
Because Reggie returned early, he is able to leave before S.T.E.V.E. makes his first trip. That, though, creates a convoluted problem. Reggie is going to travel back a few days to find Jake, and so replace S.T.E.V.E. in that encounter. S.T.E.V.E. has, in a temporal sense, not yet left from the future to seek Jake; but in a sequential sense, he already did that, and since the moment of S.T.E.V.E.’s departure has not yet been erased Reggie and S.T.E.V.E. should find that S.T.E.V.E. is already there when they find Jake. When we reach the moment of S.T.E.V.E.’s departure in this history, Reggie and Jake will be aboard and S.T.E.V.E. will not travel back to become the Great Turkey; but because we have not yet reached that moment we have not yet changed that departure, and so an empty S.T.E.V.E. will arrive to confront Jake, and be there at the same time that Reggie is trying to be the replacement turkey.
Once we reach the moment of departure, though, Reggie and Jake will be aboard S.T.E.V.E., and having already met the Great Turkey Jake will give the destination. That means that this S.T.E.V.E. will not go find Jake and give him the assignment, and we have an infinity loop, as the presence of the turkeys aboard S.T.E.V.E. prevents S.T.E.V.E. from contacting the turkeys, preventing them from being aboard S.T.E.V.E. and so causing him to contact them. Since infinity loops prevent time from advancing into the future, if Reggie has not left to find Jake by then he never will.
It thus is obvious that the story fails if Reggie and S.T.E.V.E. leave either before or after S.T.E.V.E.’s original departure time. However, S.T.E.V.E. is a smart enough machine to recognize both the problem and the solution, and he already knows the launch time (which his actions have not altered). Therefore S.T.E.V.E. must persuade Reggie to board with the intention of being the Great Turkey for Jake, then depart for the past at exactly the same moment his doppelganger does. Reggie boards a bit early, perhaps–depending on whether he picks up the pizza now or earlier after finding Jake–but S.T.E.V.E. can as easily hop forward a few hours to the departure time and then make the trip to the past.
By that arrangement, there are turkeys aboard the S.T.E.V.E. being launched from the government facility, so that S.T.E.V.E. does not look for Jake but goes directly to the past, but he is replaced by the other S.T.E.V.E. who leaves from Camp David at the same instant and makes the trip the original S.T.E.V.E. made to find Jake and give him (this time through Reggie) his mission. All of history is preserved–but for a few quirks that have to be addressed along the way.
To briefly cover four centuries of history as it has been presevered, Jake and Jenny attack the fort without Reggie, the Pilgrims use the cannons, and turkey remains on the Thanksgiving menu. This raises another serious issue. All the captured turkeys will be eaten, and most of the rest of the rafter massacred. Some of these perhaps would have become part of the Thanksgiving feast anyway, but without Jake many more would have escaped and survived. The probability of both Reggie and Jake being born dwindles. That has no impact on our present versions–time has not yet reached the moment at which they cease ever to have existed–but it is unlikely in the extreme that Reggie would see himself. That doppelganger probably does not exist.
There is yet another quirk in this.