If you could magically extract everyone's scalar utility function, then it wouldn't be a voting system. The requirement that you state is in the definition. It

If you could magically extract everyone's scalar utility function, then it wouldn't be a voting system. The requirement that you state is in the definition. It's just implicit in the word "voting" rather than explicitly stated. Arrow's Theorem does rely on an explicit definition of voting, which is that voters make discrete choices about their candidates subject to certain criteria. Some theorem (possibly part of Arrow's Theorem) says that it's not possible to eliminate the potential for strategic voting in a 3+ party system without negating other desirable qualities like those listed above.