Have you looked at POBox.com, a sister company to Fastmail? They have a forwarding service where you can do everything you want, but instead of storing your emails they are forwarded to another service like Gmail or Outlook.com. But, POBox.com handles all the domain email side of things. $20 a year.
https://www.pobox.com/pricing I've used them before. Very simple and reliable, with the Fastmail team behind it all, though customer service is handled from Philadelphia in the USA I believe.