The game already lets non-fighters train maneuvers via a feat so we are part of the way there.
Someone asked a question a little while ago about how one might allow other skills to affect combat... And it kicked off a thought i was already developing. The Fighters maneuvers represent things skills might be used for, for instance intimidation could induce any of a number of effects but specifically many of the maneuver results from the fighter kind of pop to mind (like pushing them back or making them bobble their weapon). The end result is an idea that those trained in a maneuver and a skill which might be used to implement that effect can choose to do so by spending an attack action (if they are trained in the maneuver they have superiority dice and the subsequent action gets a bonus based on (perhaps always a d6) and the attribute associated with the skill). If they are not trained in the maneuver or not trained in the skill it takes a skill check and you might still get the attribute bonus. If untrained in either well find a different appropriate skill

.
Religion or Diplomacy might enable a Rally for instance.
One of the benefits of this every extra attack benefits the fighter and they are not doing a weak attack to do the special one they are putting more eggs in a single basket to have some interesting end effect and using a secondary or tertiary attribute for it too. (at some level this might always be a fair trade)
The un-trained in the maneuver individual is actually sacrificing an attacks worth of effectiveness to make their next special its not as good of deal but sometimes might be worth it.
Which is kind of what you were talking about?
@doctorbadwolf