After all, the heads of the Ziu aren't selected with the goal of choosing a reliable and efficient administrator
The heads of the Ziu are selected with the goal of choosing people to run the Ziu. Now I don't know if this is different in the US, but in Germany, this includes setting up the order of business for parliamentary sessions, presiding over debates in the respective chambers and being in charge of parliamentary votes. In Talossa meanwhile, debates don't happen in the chamber but rather in the Hopper (if at all), parliamentary sessions dont exist outside of Living Cosăs which havent been a thing for over a decade, making orders of business pointless and the Secretary of State is in charge of vote tallying. So then, what do the Túischac'h and Mençei actually do all day? And why should it be the Secretary of State's responsibility to pamper them? The fact that there were long periods of time during which there was no Túischac'h without anyone noticing or trying to do something about it is pretty damning.
As the Túischac'h I want to be in charge of the inner workings of the Cosă and its (hypothetical) debates. What else am I supposed to do in that office?
There hasn't really been a purpose to either office, frankly. Some folks in particular have felt that those offices should do things and have responsibilities, but that's pretty new. Up until recently there's mostly been ceremonial (well, since there weren't any relevant ceremonies, maybe not even that).
I think it's great to change this. But while you are correct to note that the heads of legislatures in Germany and America are in charge of things like scheduling votes and the like, they don't actually do the bureaucratic stuff themselves. There are staff in their offices and in the employee of the legislatures that run votes and distribute bills and so on. Nancy Pelosi is Speaker of the House of Representatives because she's a gifted politician and leader, not because she's good at administering the House records database.
You guys should run stuff however it seems best to the legislature, but I just thought it was worth pointing out that usually the people who do administrative things are appointed to such posts, not elected, and that's for good reason. If you want your job to be important and have real relevance, I think that's great. I would have just thought that something more related to the actual legislation rather than administration would make more sense. Like, for example, setting a new norm that you will carefully comment on every single bill. That could be a new expectation for your job, and a new standard moving forwards (like how Menceis are now more expected to run formal confirmation hearings).
So, take it or leave it, that's just my two bence.