Having refreshed my brain, it is certainly possible to pretty much do as Weaver has suggested, I do it myself.
Most of my domains specify the same MX, directing mail to Google Apps (Now known as G Suite). Within Google Apps admin console I then add additional domains as aliases, for which Google then handle mail, after authenticating me as the owner.
This is much better than mailbox forwarding as it provides clean send/receive, with correct inward/outward addressing for each domain via IMAP. It also avoids the overhead of paying for a.n.other mail provider for each domain, to perform the forwarding, and of administering multiple different mail hosts.
I was lucky, signing up to Google Apps when it was free, a deal which they continue to honour. Since about 2012 it is no longer free, far from it, and so unlikely to be an attractive option. But if Google can do it, so can others... are there really no similar services elsewhere?