So it varied widely based on the branch of service, and I only got to look at the hard numbers for the Army Reserve, although I have it on decent authority that the Regular Army was similar.
USAR didn't disapprove any correctly applied for religious exemptions. There was a memo that came out with instructions on how to request one, and if you followed those instructions, they were either approved, or no action taken (you got to stay as pending a decision) until the mandate was lifted. Medical exemtipons were a little different, but the TLDR was if your doc agreed with you, you got the exemption. I know a couple folks that Twitter told them they should get an exemption for whatever ailment they thought they had, but the doctor's didn't agree, and it was denied. Interestingly enough, at least in the USAR (where we tend to be a little older and more broke) the process of going to the military doc and claiming all your woes to try and get a medal exemption often triggered a medical fitness board that resulted in medical discharge.
The soldiers that actually got discharged (rather than medically separated or retired) were the ones that just said "No, You can't make me", and refused to follow any of the processes to request an exemption. Partially because that was quicker, and partially because they can, in fact, make you those soldiers actually got processed out on General Discharges before the policy was paused, and later reversed. All of those that I saw were article 92 violations (Failure to obey a lawful General Order), and yes they normally included a bar to reenlistment. You are also correct that the process to remove that bar takes a little time, but not a ton. Think weeks not months in most cases.
You are also correct that the vast majority of those soldiers have no intention of coming back. They are done. There are perhaps some Regular Army folks that were close enough to retirement that coming back for a year or two makes sense, but like I said in the post you quoted, USAREC is saying very few are coming back.
Rumor mill is that the USMC and Navy were much more dickish about the whole thing, and more of them might have gotten shafted.
