You specified the aerial component as the defining component. If that's the definitive component, then we don't need a midfielder in there, just someone good at aerial duels. I was making a point that you can pick out any one component and say "Player X is better", but when you look at the qualities of the role, it becomes a different picture. Rodgers put Gerrard there to control the game from the middle of the field, not to break up play. So although he and Lucas are good at aerial duels, that component is of lesser importance than the ability to take the ball off the back four, hold possession under pressure, turn with the ball, get the ball forward, be a pivot in the middle to switch play, play first time accurate passes, get the occasional goal, and turn and recover if the ball is lost wide. Given those extra qualities, Allen comes out ahead of Lucas in the overall picture. That's not to say that Lucas is bad at them, but Allen is better, and has the added factor of being more mobile and quicker, which is a huge bonus in central midfield.
Let's cut to the chase: is there an aerial dimension to the role as Gerrard has played it? If so, it is part of the role. Its ranking relative to other aspects or dimensions of the role is a matter of debate. I didn't rank it highest, let alone exclusive (which your Glen Johnson 'counter-example' seems to indicate).
My argument is that, taking all components of the role into consideration, including aerial battle winning ability, there is no clear and unequivocal ranking of Gerrard, Allen and Lucas for the DM/controller/pace-setter role. Surely, when it comes to pace and ability to change directions quickly Allen shades both Gerrard and Lucas. In other aspects of the role (hold possession under pressure, turn with the ball, be a pivot in the middle to switch play, play first time accurate passes), Allen and Lucas are comparable to each other (functionally equivalent) and superior (slightly) to Gerrard. In the long-passing and goal-scoring aspect of the role, Gerrard is by far superior to both Lucas and Allen, although both are capable on the former, and Allen in the latter.
To me Lucas is superior to both Gerrard and Allen in recovery tackles and headers high up the pitch, even with his now more limited mobility. My freshest evidence is from the match vs CPFC at 2-3 and 3-3.