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What the post office does and gets away with and what is required by the ATF are not the same thing.
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You don't understand how government works, do you?
Believe it or not, there are many things that the government requires the use of stamps to prove that a tax was paid, but the government no longer requires people to have an actual stamp to prove tax or service paid for.
The watermark stamp on e-file forms is all you get with e-file. And lost forms that get replaced don't even get a stamp.
So the reality of it is that while the reg may say 'physical stamp' the BATFE has pretty much rendered that requirement moot, and would have an impossible task of prosecuting your for failing to have a physical stamp on a Form.
If you have a registered item, the BATFE field offices can verify it pretty easily, whether or not you can provide a physical stamp.