iBump2.0:
In the computer game, the owlery is in a separate building, which would then necessitate going down to the entrance hall in order to then go up to the owlery.
As for your main point, remember that all Dumbledore has to go on is what Harry told him. So he has no reason to think that Hermione didn’t actually send him an owl, though he missed it. He could have “crossed paths with it” quite some time before he bumped into Hermione in the entrance hall. Thus, she could have had time (in Dumbledore’s mind) to run all over the place, so her location when Dumbledore met her would not be indicative of whether she had already sent the owl. This whole timeline seems to be assuming that Dumbledore flew to the ministry and back, which Mcgonagall does say explicitly. Which begs the question – why didn’t Dumbledore just apparate?