You might be interested to know that there's a voting system for politicians which works similarly, called "Mixed-member proportional representation." 50% of the politicians are directly voted and then 50% are appointed in order to make the party distribution look more like the voters should have expected it to look. In the analogy you're proposing, "parties" become the "bureaus" of the bureaucracy. You can also imagine their "programs" becoming a sort of "party list", I suppose, and you could even do a party-list proportional election in that sense. (Imagine if the 50% was instead 100%.)
The problem is that "fixed seats" are the wrong way to look at this; the fixed quantity is actually the budget money made available by the government. So, the last budget was a $3.54 trillion budget, and we might be able to "chunkify" the budget into $20 billion "awards" and thus "elect" 177 programs in the various departments and agencies to be allocated those awards. The US has 15 executive departments which is about the limit which democratic elections on this sort of scale usually take, so that would probably be the limit. Each department would put several large "programs" in order and you could choose some programs and some departments, or so; we then fund the most popular programs, followed by the top programs in the funded departments, until the department allocation matches the national average.
The problem here is: with political parties, anybody can start one. But the executive branch starts new executive departments and presumably builds those prioritized lists. Suppose the executive branch decides that there is only one department, they could more or less set the entire budget as they pleased.
The problem is that "fixed seats" are the wrong way to look at this; the fixed quantity is actually the budget money made available by the government. So, the last budget was a $3.54 trillion budget, and we might be able to "chunkify" the budget into $20 billion "awards" and thus "elect" 177 programs in the various departments and agencies to be allocated those awards. The US has 15 executive departments which is about the limit which democratic elections on this sort of scale usually take, so that would probably be the limit. Each department would put several large "programs" in order and you could choose some programs and some departments, or so; we then fund the most popular programs, followed by the top programs in the funded departments, until the department allocation matches the national average.
The problem here is: with political parties, anybody can start one. But the executive branch starts new executive departments and presumably builds those prioritized lists. Suppose the executive branch decides that there is only one department, they could more or less set the entire budget as they pleased.