Or, perhaps, .tech is different in some way from .com in terms of
Yes, and Patrick Mevzek explained that. But the issue maybe is not the extension but the particular domain name you chose. Many of the new registries apply differential pricing on so-called premium keywords. Which means that generic keywords are going to cost more than ordinary ones ...
Most of the domain name market is open to competition, and hence you can shop around. You are not giving at least a ballpark of the current prices you are contemplating so it is difficult to know how far you are from example from registry prices.
However, in shopping around, no one else seems to offer what I would consider reasonable rates.
You don't do this in DNS. It has to be done on the server where the MX Record is pointing.
In cpanel it would be an email forwarder. If you don't have cpanel or a hosting panel, you will have to search for how to do it in your OS e.g. Centos, debian etc.