Are we sure we truly know what AATIP was all about? Yes, he could very well have worked the program on behalf of/with/for DoD. That's not the issue. The issue is the public interpretation of all this.
I agree. For what I have understood, the program was kept secret but it wasn't classified, so it doesn't sound like they were the sort of people who would have easy access to all relevant classified material etc. The reported $22 million budget means something like 300-400 000 per month, and they reportedly spent some part of it to infrastructure etc., so it's even less for the rest. Most of that went reportedly to a private company, which presumably has more restrictions in accessing classified materials. To me that sounds like there probably weren't that many persons working for that at the Pentagon, possibly only a few managers to manage subcontracted work.
We also know how Elizondo has covered his real goals by appealing to the aviation safety aspects:
Head of Pentagon’s secret ‘UFO’ office sought to make evidence publicElizondo, in an internal Pentagon memo requesting that the videos be cleared for public viewing, argued that the images could help educate pilots and improve aviation safety. But in interviews, he said his ultimate intention was to shed light on a little-known program Elizondo himself ran for seven years: a low-key Defense Department operation to collect and analyze reported UFO sightings.
So maybe that program was also represented in a similar way, which would mean they didn't have an official "UFO program", or at least it may depend from who you ask. For some it might be just some aviation safety program few people worked on.