Is the Grand Maester nominally in charge of the Citadel or is he just supposed to advise the crown? I imagine it would be difficult to supervise the entire organization from King’s Landing.

The latter. The Citadel is governed by the Conclave of Archmaesters, although they elect a Seneschal to do the day-to-day governance on a yearly basis.

To use a modern-day analogy: the Seneschal is the department chair no one wants to hold because it distracts from their research, the Conclave is the department meeting that’s full of vicious petty politics despite having very low stakes, and the Grand Maester is the guy they send to advise the government so that they don’t have to be distracted from their research.