My obligatory Python based answer.
Given the requirement not to use a data collection company (there are a lot of them out there) and that most if not all of the free SMS sending services are run by some of them you need something that can:
Check your emails
Run some filters on them (presumably you don't want spam notifications)
Send the SMS
I would ...
It seems like DetoxDroid could fit that bill. It's intended for "digital detoxing", and one of its features is a global DND mode ("mute all applications"). It lets you define exeptions ("…except one"), thus matching your requirements.
For now, it's not available at any store I know of, and APKs must be downloaded from Github and ...