Dig them out and redo them. I built a small deck on clay soil about eight years back and left two posts embedded the same way, thinking ground-contact treated lumber was tough enough. Pulled one out last spring for unrelated reasons and the bottom foot of that post was soft enough to dent with a thumb. The other two that sat on brackets above the footing were still solid as the day I set them. That was all the convincing I needed. Here's what's working against you. The end grain of a post is basically a bundle of straws, and concrete sitting in wet clay never really dries. Water pools in that bowl you're describing, soaks through the concrete, and wicks straight up the end grain. Ground-contact treatment helps, but it's not designed for permanent saturation on all sides with no chance to dry out. That setup is exactly how fence posts rot, and fence posts are expendable. Your deck posts are holding up a structure you stand on. The fix isn't complicated. Pour the footing so it comes up a few inches above grade and slope the top so water runs off instead of pooling. Set an standoff bracket in the wet concrete or bolt one down after it cures. Those brackets hold the wood an inch or so above the concrete so the end grain never touches it, and air can move around the bottom of the post. Total cost for the hardware is small compared to the digging you're already about to do.
And yeah, the digging is the annoying part. But you're framing in days, not weeks. Once the beam and decking are on, pulling a post goes from a two-hour job with an auger to a demolition project. Doing it now is cheap insurance.
Your buddy isn't wrong that people have done it this way forever. Some of those decks are still standing. But plenty of others have posts that look like soggy cheese at the ground line, and clay soil is about the worst case for it. Redo the two holes, keep the other four if any of them were done right, and you'll never think about it again.