Almost all (old) providers let you set up a mail forwarding (to the new provider). But even better, several services let you set up some fetchmail-like program where it imports all the messages from the old provider, e.g. via POP3 or IMAP, and even deletes them with the old provider, if you want.
Nevertheless, I fully agree with you that email services are much too expensive. Given the current market prices, Fastmail, for example, is great and worth its money.
We're told to spend some money on good quality software and services.
But: If I (and even more so the average non-technical user) pay 99 USD/year for Dropbox, 50 USD/year for Fastmail (because you want your own domain), 10 USD/year for the actual domain (separately), 100 USD/year for my quality newspaper subscription, etc. -- and all this is just for one person in my famile (i.e. me) -- you soon end up not having enough money for the "real" things in life.