That can be considered an intentional plot hole that served as a plot device.
Though someone could argue that James trusted his friends completely and would not question their ability to become Secret-Keepers, there is no satisfying answer as to why he himself did not become the Secret-Keeper. If he was the one that held the secret of their location, then Voldemort wouldn't find them, would never succeeded in murdering Lilly, who provided the protection to Harry, which ultimately led to his own defeat.
Even so, James knew that Voldemort was hell-bent to find and kill them, so he would acknowledge the fact that making someone of his friends the Secret-Keeper, put him in a grave danger. This might be an intentional plot hole that drives the entire series.
1. James Potter could not simultaneously live in the Fidelius Charm's field and be the Secret-Keeper.
Wrong. As others have suggested in comments above, when the secret itself is the location of a building, the Fidelius Charm does not prohibit a resident of it to be the Secret-Keeper, otherwise neither Arthur nor Bill Weasley could be the Secret-Keepers of their houses as shown in Harry Potter and the Deathly Hallows.
2. James and his friends where the only ones that could be Secret-Keeper-"eligible".
Wrong. One perfect candidate was Albus Dumbledore. There is no indication as to why they didn't make him the Secret-Keeper either, which would make the secret if not unbreakable, then extremely more hard to break than the situation that we know. He did gave them this option, however, which brings us to:
3. It seems that the protection of Potters might not meant to be all that powerful enough, considering what went wrong.
Maybe. Although there is not direct indication as to whether the Fidelius Charm was the only enhancement placed on their house (the Order of the Phoenix might had provided even more powerful charms which shielded their residence), it seems that Albus Dumbledore might "wanted" the situation to be that way. By that, I don't mean that he wanted the Potters to be murdered. But, considering that he was the only one -at that time- that heard the entire prophecy (and knew that Harry would be Voldemort's only match) as also as that he is Albus Dumbledore (which means that he had foreseen the future events), he might wanted to "step aside" and leave the events to naturally flow. He would of course have provided the Order's best protection but no one can argue that he seemed hesitant to advice James that being the Secret-Keeper would be much safer than the alternatives.
Note: Dumbledore proposed the idea to be their SK but James and Lilly quickly withdraw it. Now, I know that he wouldn't have taken extreme measures to persuade them but considering the fact that we were at the peak of the First Wizarding War, he seemed reluctant in doing so.
1. Wormtail would ultimately become the Secret-Keeper, no matter which theory makes sense.
As Peter was with Voldemort's side at that time, he may posed extreme pressure on James and Sirius to change the Secret-Keeper, providing all the good reasons he could come up with in order to achieve his goal. Though we know that this would make perfect sense if Sirius was not the SK in the first place, it does bring us to the conclusion that even in that situation (or the one that we are discussing in this thread), Wormtail would ultimately become the SK, regardless of what theory makes sense or not, thus making every conversation on that matter irrelevant.
2. James wanted Sirius for no apparent explanation.
James just wanted to make his best friend the Secret-Keeper because it seemed to him that this was right. I can't possibly dive into his personality but that would not be an irrational decision to begin with.