Albus Dumbledore. Sirius Black. Peter Pettigrew.
Notice two names conspiciously missing from that list of people considered as candiates for the Potters' Secret Keeper.
James Potter and Lily Potter.
Why couldn't James have just been his own Secret Keeper the way Bill was for himself and Fleur? Or Arthur Weasley for Aunt Muriel's, when they hid there?
Besides, of course, the way that the entire series would have collapsed like a house of cards without that.
The irony is that it could have been fixed with one simple switch: Bill and Arthur could have changed duties, so Bill was Secret Keeper for Aunt Muriel's, and Arthur for the Shell Cottage. Since Dumbledore did not live at Sirius Black's, that would have made an invariable rule that a location can only be protected by a Secret Keeper who is not resident there.