In my perspective, the answer is that in USCF the arbiter should NEVER start a time-forfeit claim, neither for sudden death time controls nor non-sudden death time controls, at 4/9/2022.
You are correct. This arises because of the bizarre USCF rules regarding flag falls not resulting in loss of the game if the opponent has not been keeping score correctly. Hence in USCF tournaments the player must "initiate a time-forfeit claim" rather than just point out the flag fall.
If the opponent has an incomplete scoresheet it can be in the interests of the player whose flag has fallen to call the flag fall before the opponent writes down the missing moves. For the arbiter to intervene in such a situation could be seen as benefitting one player or the other.
FIDE arbiters are required to check that players maintain a complete up to date scoresheet and penalize them if they don't.
Here are the FIDE references:
Regarding who may call flag-fall
6.8 A flag is considered to have fallen when the arbiter observes the fact or when either player has made a valid claim to that effect.
Regarding writing the moves:
8.1.1 In the course of play each player is required to record his own moves and those of his opponent in the correct manner, move after move, as clearly and legibly as possible
12.1 The arbiter shall see that the Laws of Chess are observed.
So, writing the moves is one of the "Laws of Chess" and ensuring that these laws are observed is the arbiter's first duty.