No, your emails are still sent from your providers servers and are signed by Proton/Tuta/Gmail/whatever you choose to use. Your domain is the from: field but it was sent by the actual email provider with all the trust behind it.

I never ran into something requiring a Google/Apple address. Some may require a Google account, but you can create a Google account with any email address. You don't actually need to use Gmail on your Google account.