They can, and that's actually what they're doing, or at least trying to, here.
The key issue is that the SFUSD board are elected officials, and the superintendent is selected by them, so the mayor and supervisors can't summarily fire them.
Members of the San Francisco Unified Board of Education are elected at large to four-year terms on a staggered basis every even-numbered year.
Until the next election, the school board has the power to do what they think is best, so long as it is in accordance with the law. That's where the lawsuit comes in:
State law (passed by the legislature and governor) already requires schools to make a clear plan to reopen. The school board claims they're complying, but the city alleges that they are defying the order by dragging their feet:
City Attorney Dennis Herrera, with the blessing of Mayor London Breed, plans to sue the San Francisco Board of Education and the San Francisco Unified School District for violating a state law compelling districts to adopt a clear plan during the COVID-19 pandemic describing actions they “will take to offer classroom-based instruction whenever possible.”
When there's a conflict between different parts of government which are not directly subordinate to each other (the article says the school district is overseen by the state), the normal action is to go to the courts, to determine what the law requires and what steps are needed:
The city attorney will file a motion Feb. 11 asking the San Francisco Superior Court to issue an emergency order compelling the district to formulate a reopening plan with the aim, he said in an interview, of “getting kids back in school as quickly as possible.”
The question for the courts is whether the SFUSD's cautious reopening plan (or lack thereof) is in violation of the law pushing for reopening and, if so, what remedies are required.