I don't want a government entity, or really any entity I'm not paying directly for their services, to be the gatekeeper between me and my accounts.

The social engineering attack surface of my account currently consists of a handful of support contacts at my ISP, who have been trained to deal with computer security. If you allow any USPS employee to access your account, you've suddenly increased the potential attack surface by several orders of magnitude.

(1) Your USPS-provided email should neither be mandatory nor the only acceptable email. It could be an extra convenience, and a low-friction way to get a reasonably secure email for first-time / technically unsavvy users.

(2) An entity you're directly paying to may go out of business, sometimes due to circumstances beyond their control. At least state-sponsored entities don't do it so abruptly in most of the "civilized world".

The account recovery process for commercial email providers usually involves you photographing your ID. Presumably the post office, in person, would be far less likely to be fooled.