Should an USCF arbiter correct illegal moves without a player claim?
What does FIDE require from their arbiters?
Under the FIDE Laws arbiters do not correct illegal moves (except in one special case) but they do flag them, require the player to correct them and punish the player.
In standard time control chess:
7.5.1 An illegal move is completed once the player has pressed his clock. If during a game it is found that an illegal move has been completed, the position immediately before the irregularity shall be reinstated. If the position immediately before the irregularity cannot be determined, the game shall continue from the last identifiable position prior to the irregularity. Articles 4.3 and 4.7 apply to the move replacing the illegal move. The game shall then continue from this reinstated position.
This basically says that if the arbiter knows an illegal move has been made, whether spotted by the arbiter, flagged by one of the players or even by a spectator to the arbiter, then the arbiter must act, require the move to be corrected to a legal one and punish the player.
The definition of "illegal move" is also very broad. It includes pieces moving in a way other than specified by the rules, moves that fail to get the king out of check or put the king into check, but also includes these:
7.5.2 If the player has moved a pawn to the furthest distant rank, pressed the clock, but not replaced the pawn with a new piece, the move is illegal. The pawn shall be replaced by a queen of the same colour as the pawn.
Only in this case does the arbiter correct the move played
7.5.3 If the player presses the clock without making a move, it shall be considered and penalized as if an illegal move.
7.5.4 If a player uses two hands to make a single move (for example in case of castling, capturing or promotion) and pressed the clock, it shall be considered and penalized as if an illegal move.
For blitz and rapid the rules are different, partly because the moves are not recorded and partly because a sequence of moves can be made very quickly making intervention difficult.
A.4.2 If the arbiter observes an action taken under Article 7.5.1, 7.5.2, 7.5.3 or 7.5.4, he shall act according to Article 7.5.5, provided the opponent has not made his next move. If the arbiter does not intervene, the opponent is entitled to claim, provided the opponent has not made his next move. If the opponent does not claim and the arbiter does not intervene, the illegal move shall stand and the game shall continue. Once the opponent has made his next move, an illegal move cannot be corrected unless this is agreed by the players without intervention of the arbiter.
So, the arbiter may only intervene if no more moves have been played. If no more moves have been played then the arbiter should intervene but is not at fault if they are too slow or miss the move.
Of course where illegal moves are made and not corrected this leaves the possibility of illegal positions arising which are difficult or impossible to correct. Hence the next rule for rapid and blitz:
A.4.4 If the arbiter observes both kings are in check, or a pawn on the rank furthest from its starting position, he shall wait until the next move is completed. Then, if an illegal position is still on the board, he shall declare the game drawn.