If the Potters had only gone into hiding recently there is no inconsistency. The letter was thanking Sirius for Harry's present, and his birthday is well before Halloween. No one has to have seen the Potters except maybe Dumbledore, who had seen the cloak "a few days previously", but Lily's letter says "Dumbledore's still got his Invisibility cloak, so no chance of little excursions". If the letter was indeed sent as a thank you note for Harry's birthday then Dumbledore had the cloak for a long time, and the inconsistency lies there. Petunia sent the awful vase for Christmas, but it doesn't say which one.
The house itself was also hidden. Harry's POV "He could see it; the Fidelius Charm must have died with James and Lily." Now, Harry could be mistaken, but that seems to me to serve no purpose to the story.
I think the most inconsistent thing is Lily's letter. She talks about Harry's Birthday as if it just passed, but she and James are in hiding. Perhaps they were in hiding before the Charm, just plain hiding? Voldy should have been able to find them then though. Maybe some other enchantments before the Fidelius Charm?
The Fidelius Charm also bothers me when the kids use Voldy's name and nothing happens. The Death Eaters show up, but still can't see the house. So the Taboo breaks all protective enchantments except that one?
Anyway, if the house were hidden but the Potters were supposed to stay there, it would make sense to trust someone besides one of them as Secret Keeper. Otherwise if they stayed in the house no one could ever find them in an emergency. I believe they were hidden to all but one, and they ensured their safety even more in their minds by making the less obvious choice. This of course cost them dearly.