At some point I thought that it absurd on purpose but I had some people explaining me the rationale behind it (there is no rationale - if there is an exception it at best weakens the "rule")
https://en.wiktionary.org/wiki/exception_that_proves_the_rul...
Just as you say, the point is that a rule is implied by a specific exception, as in the example "free entry on Sundays", which implies the unstated rule "pay for entry on other days".
The exception weakens the rule, it's true, but may also reveal the rule.
(if there were no rule, there wouldn’t be any exceptions to it, or nothing would seem exceptional with respect to the (previously unstated) rule)