...and why would anyone see your destination e-mail if you simply reconfigured your mail client to use the other address?
Gmail doesn't allow me to do this - it'll change the reply-to email, but not the one that shows up next to my name. And stupid Juno doesn't let you forward your mail, and doesn't provide pop or imap access so you're forced to use their crappy, ad-laden software. :~) (Otherwise I would have been doing this for years already, and I wouldn't have my current problem.)
I could just forward all addresses separately to a mail client, but it's really nice to have them all in a central location online as well as on my computer, and gmail does that archiving.