I am trying to understand your position. Do you believe just city dollars pay for a "municipal" stadium, or the suburbs as well?
What if the suburbs are in different states (e.g., New Jersey communities near Philadelphia and New York City and MD and VA suburbs near DC) and thus are likely to have neither fiscal nor operational control over the stadium? Should they still have to support it with tax dollars (as opposed to / in addition to ticket sales)?
What if the stadium is not in the city but instead in is a suburb, as I think still occurs in the Bloomington stadium outside Minneapolis? Who do you think should pay for and control and operate the stadium then?
You are asking some questions related to the specifics of some given situations, where I am responding to the general question:
"Should tax dollars be used to finance Municipal Stadiums (baseball or football)?"
My answer to that question is: Yes. It can be a reasonable use of tax dollars to construct such a facility.
As to the specifics of who pays for what and suburbs vs. city or county, well, as the saying goes: "The devil is in the details."
But, I would say if a benefit can be determined, for a certain city, by the construction of a near by stadium in differeing city, then yes, there should be shared costs, profits and decisions.
Case in point: The LA Memorial Coliseum Commission. This entity runs the Coliseum and Sports Arena, which over the years has been home to the: Rams, Dodgers, UCLA, USC, Lakers, Kings and Clippers. It is run jointly by a committee of people appointed by the state, county and city.
They have also proved to be about the worst run agency of it's type in history. They have succeeded in driving off each of their major tenants, except for USC. However, the fact that it is a poorly run entity does not take away from the fact that there can be a system for shared revenues and costs.