I wish it was that easy to be sent home. I'd wear flip flops every Friday.
I know that you mean this as a bit of a joke, but it is actually a good question to ask in looking to resolve OP's three original questions. If you actually did wear flip-flops in on a Friday and told your manager that they are the only shoes you have, then of course they would not work you.
The company standard for "proper footwear" is a sturdy leather (or leather like) upper, and a nonslip sole. It is documented in the 30 day training packs for hub, preload and driving positions.
We don't have enough information to answer OP's first question. It may be that the manager handled it properly from a safety/labor standpoint, but not an employee standpoint. It is likely that the manager actually did put everyone "on notice" about proper footwear. At a minimum he would have been able to demonstrate that the employees knew what proper footwear was from their training documentation. Just because you don't remember signing that page doesn't mean that you can claim ignorance from a labor standpoint.
Are the grievances winnable? Probably not. The manager would have been wise to also document not only who he was turning away, but to be taking pictures of what their footwear was in the event that it did go to a panel. Again, if we can agree that he would not work you in flip-flops then the case could be made that he should not work you in canvas sneakers either. There is an argument that could be made that this training/policy on proper footwear meets the definition of an "extracontractual" policy, which is expressly forbidden in the national master. There was a BA in my local that had this as his personal mission. I never saw him get any traction with it.
Does management have to abide by this rule? Yes absolutely. Specifically I'm referring to operations management. If the district staff is in your office, then there is a different policy regarding the footwear that they must wear. Remember though, the actual standard is not boots. I wore the same style of Rockports for years and the met the standard.
It is always interesting to me that employees/members want to fight this policy. As a former manager, there are actually not many things that I will defend the company on. However, the changes that I saw over 15 years to the safety culture it's something that I will defend. Is it really where it needs to be? No. Is it perfect? No of course not. Seeing how safety is (or is not) a part of the conversation at other companies now, I will say that UPS is light years ahead of the broader American industrial culture. I'm not kidding when I say that there are companies out there that still look like a 1920s Chicago meet district processing plant when it comes to safety.
Edit: for what it's worth, the best way to win this grievance would have been to document what all of the managers, full-time and part-time supervisors were wearing that day. If any of them were allowed to work in footwear they did not meet the same standard that hourly employees were held to, then the grievance would be winnable.