I reckon a player's involvement - specifically how often they should run the ball - is more about the gameplan than the player's whim. It is also clear that we don't always have two players on the field who are acting as 'props', even if we have three players on the field who are capable of playing such a role. Mostly we have one to take the first hit-up, and two others to support or take the next run as necessary plus defend in the middle with the 'prop' and the hooker. Plus sometimes one of the middle defenders is Joseph Paulo, who plays 'middle half'.
Wait - are you suggesting teams have gameplans??
Next thing you'll be telling us the waterboy is reminding them who should be doing what...
I hate the term 'middle half' - it very poorly reflects what Paulo (and others) is doing. His main job is NOT to think about passing, or to get in position to kick, or to organise a play. His main job is meterage through the middle, however he is clearly told that he can play the ball before the line, through the middle third of the field.
Glenn Stewart is not the 'right half' for Manly.
Greg Bird is not the 'left half' for Gold Coast.
Lewis Brown is not the 'left half' for Penrith
James Graham is not the 'middle half' for the Bulldogs
Corey Parker is not the 'middle half' for the Broncos
These guys are just skillful forwards whom the coaches have written plays for in the playbook.
I need some maths to make my point: 1/2 + 1/2 = 1. Therefore there can only be 2 halves on the field.
Aha, you say, but a half is defined by their role. You fell into my trap, that's precisely my point. If you watch all teams play footy, watch what halves do, then carefully watch Paulo, his role is nothing like that of a half. He is not there to kick, nor is he there to run the sweeping pass plays at first receiver, he is a 'runner' of the football who is encouraged to use his extra skills to open up the middle of the field for the forwards.
If you wanted a 'best' description, then you could use the term 'standing hooker' - because his role is most similar to that of a dummy half (not a real half, a pretend one by definition!) in that he is looking to play the forwards onto the front foot, or defer to the 'real' halves.
I make the distinction because IMO I have seen a 'middle half' tactic actually used, in fact we all have. SK used Casey McGuire in the role for the first part of a season. It was disastrous, and stupid, and never to be repeated.