* Increased bureaucracy (why not add a 0.1% to an existing tax?)
* Regressive taxation (you pay the same amount regardless of your income; if you used the income tax, you would distribute it more fairly)
* No market shaping (you could instead fund it with a tax on, say, TV ads for high-calories sodas, or whatever as a society you decide should not be forbidden but discouraged)
What stops legislators from adjusting the fee at will? There’s legislation enacting the fee, defining the fee, enforcing the fee. Surely, there can be legislation to modify the fee and remove the fee. Same as any other tax.
The BBC is dependent on a fee collected by the government, as regulated by laws the government has passed and the government enforces. Insisting it's not government funded is trying to invent a distinction where none exists.
Howerver I guess it depends on the country you're from.
So in order to change the fee, it is not just one govnment that can do something, but a grand total of 17 (federal plus 16 states). Usually they all agreed, but with different ruling coalitions in each state, this mix of left and right having to agree in itself prevents the public bradcasters from becoming a tool for government propaganda. They do represent the establishment, so. Populists hate this.
EDIT: To be really accurate, the legislative branch sets the fees, not the executive on. The legislative consists of two houses, Bundestag and Bundesrat. The Bundesrat is controlled by the state governments, which are composed of local arms of the parties present in the Bundestag (kind of like congress) and regional parties, mostly those state and federal governments are coalitions of multiple parties. It is the party line thing where it gets blurred between executive and legislative bodies.
EDIT: I fell for a common misconception, as an other user pointed out. It s not the Bundestag and Bundesrat deciding these fees, but rather a compact of the 16 states. The same states make up the Bundesrat, the compact is not the same body, so. Which further decreases the influence the federal executive, and even legaslative, branches have o public broadcasting.