What qualifications does a "scrum master" have that a software engineer does not have when it comes to creating a self-managing team? The "scrum" pamphlet can be read on a lunch break and I don't think the certs even have exams.
The team members are too busy with agile meetings to write software. That's a more common problem.
Maybe giving the tech lead a technical secretary is a better idea if you have a lot of administration to be done.