This isn't entirely correct. It isn't a one-to-one mapping. You can give the same hidden email to multiple people/sites...and if they email you, you can reply to them using the same hidden email address.
I actually did this several months ago when I was looking for something that was out of stock on multiple sites. I went into Hide My Email settings, created one hidden email, and used that to sign up for in-stock alerts on those sites. I got emails/newsletters from all these sites, but once it was in stock somewhere, I just deactivated that email address...and boom, no more emails from any of those sites anymore. Anyway, that is a bit off topic and doesn't exactly help with your dilemma.
I did tinker with this and found a way to get it to do what you want. It's a bit confusing and YMMV.
Sent the following email:
To: My Hidden Email
From: My outlook.com address
After I received the email to my hidden address (which is set to forward to my outlook.com address), I tapped the leave@company link and it retained the Hidden Email as the from address. I sent the email, and it behaved as expected. It sent to that new address using my existing hidden one.
Now when I tried this using my iCloud address (and changed my forwarding to there), it didn't work...
To: My Hidden Email
From: My iCloud.com address
Tapping the email link put my iCloud email as the from address. I'm not sure why using my Outlook address worked, but my iCloud one didn't. I should also note that I did this from my phone (iOS 16).
Otherwise, I wouldn't worry too much about disabling the address (which I don't consider to be rude). It's pretty much one of the purposes of this feature...so you can end communication on your terms in case the other party doesn't comply with your requests to opt out. The email will bounce back to them and they'll see it couldn't be delivered. It could even be all automated on their end to handle bounce backs and remove it from the list.