You're right that he could have signed with any team this summer, though he'd have had to give notice to the NHL office that he had ceased to be a bona fide college student.
The rules are a little complicated but essentially when someone is drafted at age 18 or 19, receives a bona fide offer from the team that drafted him and then becomes a bona fide college player prior to the June 1 two years after he was drafted, then the drafting team retains exclusive rights until the player's college class graduates.
If the player doesn't remain a bona fide college player and gives notice of that to the NHL league office then the drafting team retains exclusive rights until the 4th June 1st following his draft (or August 15 if he was in his 4th year in college and either used up his college eligibility or was scheduled to graduate that year.)
So to sign elsewhere, Moore would have to give notice to the league office that he is no longer a bona fide college player, wait until June 1 (or at least another 30 days if he gave notice later than May 1) and then he'd be eligible to sign as a free agent.
Anyone who wants to slog through the language of the collective bargaining agreement can download it from the Players' Association website at
Collective Bargaining Agreement
The relevant provisions are in Article 8.6(c)(iii) and (iv).