[shoo_pl]

Not really, this only works for other emails hosted by Gmail (including Workspaces) or if you supply SMPT that will send those emails. If you use simple email forwarding from your DNS provider, you don't have SMPT server to give to gmail:/

[phi0]

Google will happily send from smtp.gmail.com, after verifying that you own that email. You won’t get DKIM, but Google’s reputation is enough to make the mail land in people’s inboxes.