Waterfall-style projects had 3 to 6 month timeframes of delivery and we all know how that goes. The result is always either out of date due to changing requirements or not what the customer wanted because there is no way to change course after the project specification was locked.
...or you spend so much time doing an exact specification that the Agile team has already delivered 3 incremental versions.
Waterfall were 1 to 2 year projects with heavy up-front administration. 3 to 6 months is well-balanced and that's what the "elite" companies and research groups tend to use. Two weeks is the current fad at non-tech enterprise.
After the team is mature enough, one of them can take the SM duties in addition to other tasks because the actual process runs without extra management.
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.
The real solution is of course not to have any "sprints" at all. A manageable chunk of work is a 3 to 6 month project.
Non-tech enterprise treat software engineers as children and create bad software. FAANG and adjacent companies treat software engineers more as professionals and create better software.