The FIDE position starts with occasions when the player does not have to write the moves. There are two such occasions:
- When disability, religious belief or ignorance (beginners cannot be expected to know how to write the moves) prevent a player from writing the moves. In this case the player can have an "assistant" (may be a friend, may be someone helping the arbiter) write the moves down for them. Here is what the rules say:
8.1.6 If a player is unable to keep score, an assistant, who must be acceptable to the arbiter, may be provided by the player to record the moves. His clock shall be adjusted by the arbiter in an equitable way. This adjustment of the clock shall not apply to a player with a disability.
If no such "assistant" can be found then the moves will just not be written down for the player. The arbiter is not going to suspend their duties to sit down and record the moves.
- In cases where there is an intermediate time control with either no increment or increment of less than 30 seconds. This almost never happens in the modern age. Occasionally there will be a top level tournament with 60 moves in 2 hours, no increment, then 1 hour with a 30 second increment to finish the game. The lack of an increment is supposed to make the game more exciting for spectators.
Here the rules say this:
8.4 If a player has less than five minutes left on his clock at some stage in a period and does not have additional time of 30 seconds or more added with each move, then for the remainder of the period he is not obliged to meet the requirements of Article 8.1.1.
8.5.1 If neither player keeps score under Article 8.4, the arbiter or an assistant should try to be present and keep score. In this case, immediately after a flag has fallen the arbiter shall stop the chessclock. Then both players shall update their scoresheets, using the arbiter’s or the opponent’s scoresheet
So, in this case the arbiter or an assistant should "try" to be present and keep score but it is not a requirement. Tournaments where this might occur are usually elite tournament with digital boards recording the moves. Realistically the arbiter should be watching as the time pressure is more likely to generate events which require their intervention but writing the moves down is pretty much a waste of time for the arbiter.
The USCF rules are similar, just expressed differently. Here is what they say:
15A1. Players unable to keep score.
a. Players determined by the director to be unable to keep score due to physical handicaps may have assistance in scorekeeping as described in 35F, Rules for visually impaired and disabled players, and should be excused from scorekeeping if such assistance is unavailable.
b. Players determined by the director to be unable to keep score for religious reasons may be excused from scorekeeping or permitted to have assistance as in 35F, Rules for visually impaired and disabled players, at the director’s discretion.
c. Beginners who have not learned to keep score may be excused from scorekeeping, at the director’s discretion.
So, as in the case of the FIDE rules the player who cannot for whatever reason keep score can have an assistant to do this for them.
15B. Scorekeeping in time pressure, non-sudden death time control.
If either player has less than five minutes remaining in a non-sudden death time control and does not have additional time (increment) of 30 seconds or more added with each move, both players are excused from the obligation to keep score until the end of the time control period. Doing so, however, may make it impossible to claim a draw by triple occurrence of position (14C) or the 50-move rule (14F) or a win on time forfeit (13C). Scorekeeping by both players must resume with the start of the next time control period, and missing moves should be filled in (15F).
So, no mention of either arbiter or assistant recording the moves for the players. There then follows several sections on players bringing their scoresheets up to date when the time control is reached all with no mention of a master scoresheet maintained by the arbiter. Clearly the arbiter is not expected (nor allowed) to record the moves on the players' behalf.
In general the arbiter is not required to record the moves for the players at any stage of the game for either USCF or FIDE. For FIDE the arbiter could consider writing the moves in the rare case of a tournament with an intermediate time control with no (or short) increment but is under no obligation. In elite tournaments with a chief arbiter, one or more deputy chief arbiters and several ordinary arbiters it might make sense for one of the lower arbiters to be appointed to this task.
If there is only one arbiter and several games going on then it makes no sense for the arbiter to sit down at one board and record the moves. They are supposed to be observing all the games. If there is only one game left and neither player is recording then it may make sense.