Hmm, I transitioned from Gmail to my own domain years ago, but it's still Gmail. However, I'm still both receiving and sending mail from my old @gmail.com address, using POP. Do you know if that's still possible?
(I don't actually know how often I send email from that address, so maybe I don't even need that, but just in case.)
I set it up a long time ago this way and I'm unsure of I had to somehow configure Gmail to be able to send from my other account. But I have the same also for @live.com and @zoho.com as I was trying different providers. This allows me to easily reply to forwarded emails with the old email addresses.
Edit: I think you got with the alias, didn't know this is what is called in Google https://support.google.com/mail/answer/22370?hl=en#zippy=%2C...