If an official allows a player into the game after they have legally reported, is this ruling correct?

Allowing a player to enter the game after they have legally reported is indeed the correct procedure in basketball officiating. When a player reports to the official to check in, it indicates that they are ready and eligible to play. The process follows the rules that govern substitutions, which require that players report to the official before they can enter the game. If the substitution is made according to the regulations—meaning the player has completed the reporting process and the play is stopped or within the proper timing to allow substitutions—then the official's decision to let the player in is valid. This promotes fair play and adheres to the established rules of the game concerning player substitutions.