My understanding is that in our troop only the SM, ASMs, and the JASMs can sign off on requirements. Since the JASM hasn't gone on any of the activities (he does regularly attend the meetings) since my son has joined, he has less of a relationship with him than he does with the SM and ASMs.
With regards to only leaders signing off on requirements being indicative of a troop run by the troop method as opposed to the patrol method, I would say that our troop is a hybrid. We have strong patrol identies and use new scout patrols that stay together for the duration. On campouts, we use the troop method (one reason is because there is not enough equipment for each patrol). This appears to be changing to be more patrol method as the troop is going through significant growth. I would say that while we are hybrid when it comes to the patrol method, we are more boy led than leader led (though there could be improvements here too).
After reading a thread about over zealous parents, I took a long hard look in the mirror to make sure that wasn't me. I do want my son to achieve first class so that he can start participating in some of the "high adventure" activities that are reserved for those of first class and above in our troop. Beyond that, it is entirely up to him to take the initiative as it comes to merit badges and advancement. Last night, I asked my son if one of his friends had a bike accident, would he know what to do. His answer was, depending on what was wrong, he felt that he could apply what he has learned. So clearly, he is ready to demonstrate his skills to the leaders to show what he has learned.