In case you were unaware of this discussion. It started here. Progressed to this posting on the ship Angel Gabriel. Then came this posting, and finally this posting. You must read all the comments because it was a back and forth discussion.
My final comments are my usual when it comes to genealogy. It is up to the person to make up her or his mind as to how to interpret the evidence. No one can prove the Burnham were on the Angel Gabriel and no one can prove they weren't on the ship. It is solely up to you to decide how you want to proceed. I've always been a big believer that you can put whatever you want on your genealogical charts. It's only when you put those charts in the public arena, that you need to prepared to defend your research.
I would put more credence in this story if there were true family lore involved. However, the family lore seems to be of the 19th century from sources in print. I truly believe the Burnhams were placed on the ship because people thought Robert Andrews was on the ship, which we now can prove was untrue. Certainly they were in New England by 1637 at the latest judging by the Pequot War service. If you didn't see this, it's a picture of the grave of Mary (Lawrence) Burnham, wife of Thomas.