I'm going to assume that given the details of what you did look at, you did not look at the resource stockpile. Without resources, your automated sector is unable to perform build actions even if it wants to.
When you create an automated sector, you need to either give them resources (minerals and energy) directly, or put it in the shared stockpile pool which all automated sectors have access to. Automated sectors will only take from the shared stockpile when they don't have a personal stockpile.
The stockpiles can be accessed via the Planets & Sectors menu. The shared stockpile is at the top of the list, the sector-specific stockpiles are in the sector list.
Clicking this button is a frustrating experience, it feels like you just hired someone to twiddle his thumbs.
Welcome to government.
What I imagine should happen is it should start building/replacing buildings or maybe the same for districs (I just recently found out that there were also buildable districs, the UI hides this from us it's poor UI-design) or maybe (ideally!) both.
This is what the sector automation actually does, once it has the resources to do so.
Why not even auto-hire leaders if they die?
Because you can't reasonably expect the AI to make the right trait decision for you. There's also other considerations such as the age of the applicants.
Why not even auto-generate a sector and set it's mode when that becomes available?
Because sectors that build sectors are liable to recursion issues. There are many non-trivial considerations here than a simple automation script cannot handle.
How do you expect a sector to choose its own focus? Should it pick its focus based on what it can output the most of (modifiers etc)? Or should it pick its focus based on what your empire is lacking in?
Whichever option you choose, any player who would prefer the other option would be endlessly annoyed by having the sectors make the wrong decision for themselves. Therefore, it's left up to the player to make key decisions that are much too complex for an algorithm to do in a way that everyone agrees with the decision logic.
Also, sectors do mostly auto-generate since the latest update. A sector is defined by its first planet and automatically includes all other planets within 4 jumps from it.
Also the button is just as effectively useless if the reason it does this is that it only takes effect after several years. There's an economy to be balanced and this is a matter that bears urgency.
Usually, you shouldn't be automating sectors until you're at the point of having a stable economy. If you're still at the point where you need to keep a fine balance, the AI is never going to do an adequate job at that.
Automation becomes practical when you no longer need to squeeze the blood from a stone and are just hoping to lessen the workload (by not having to click it all yourself) at the cost of some efficiency (the AI is never perfect).