They have this ability regardless of the structure, they are the owners.
Separately ,where does these cheap accusations come from?(I'm not saying they aren't) They tried to help finance a new stadium in Manhattan which if it wasn't for politics had a very good possibility of happening. From what I have read they have state of the art training and medical facilities. Idzik was brought in to be an axe man (never should have had control of the draft obviously) to try and clean up Tannenbaums mess to give the team some of the flexibility they have now.
While Mac should've got the knife earlier, he and Heimerdinger had a better understanding of the teams needs than anyone they could have brought in so I can kind of understand the thinking of letting them run the draft and as far as I know we don't know who had final say over the picks. Mac also did a good job with trades and somewhat not totally screwing them up with horrible contracts.
As it stands now some people are getting in a tizzy over Twitter rumors. It does seem someone is trying to use the media for negotiations or stir up sh*t. The media is also bored and they all want to be the guy/gal to come up with the next clickbait scoop.
As far as the titles go it's not like they are promoting someone over someone within the same organization and even if they were if they consider the person the better candidate so be it. After all they are replacing someone with a higher title as it is.
After all is said and done one of my wishes is other teams will look at the Jets to raid for coaches (except for Jim Bob Cooter ,love the name not the arrests) and front office personnel. While the Jets could get some stability and promote from within.This is one thing I admire about Pittsburgh.