More people definitely need to be aware of this; the fact that Time Machine doesn't make this super obvious to the user needs to be corrected by Apple since it would be very frustrating if you relied on it and had to discover this the hard way (as you, unfortunately, did). However, I disagree with your assessment that this is "broken". Here's my reasoning:

I think this actually works "as intended". Since keychain items serve as the root for your credentials to various services, it makes sense to protect them with two factors; in this case it's "something I know" (password) with "something I have" (device). A loss of either should render the encrypted data useless.

Local keychain is precisely that; local. It is not intended to be something that is by default transferable, as opposed to say, iCloud Keychain.

I emphasise by default because yes, a user should be allowed to easily export/backup the local keychain if they want. The fact that its a PITA and requires a workaround via AppleScripts is very frustrating. And, as I said, lack of clarity by Apple/Time Machine of this fact.