After thinking some more about it, I think I am starting to understand the logic. As things stand we have midfielders of roughly 3 types:
1. Holding midfielder (anchor): Can (1st choice), Lucas and Stewart (back-ups). This is a player who mostly holds their position in front of a back 4 and whose main job is to break up attacks.
2. Roaming/passing midfielder: Henderson (1st choice when fit), Milner and Brannagan (back-ups). This is the closest thing to a playmaker, but in Klopp's system this player also needs to be an energetic runner. I think Dahoud would fall into this category as well.
3. Attacking CM: Allen* and Grujic. Obviously this is where Wijnaldum fits in. Zielinski was supposed to fill the same need. Last season Milner was often used in that role, with mixed success (good assist numbers; not enough goal-scoring threat).
*Allen is an interesting case, because apparently Rodgers and Klopp have completely different perspectives on him as a player. Rodgers used Allen primarily as a passing metronome (category #2) or out of necessity as a holding player (category #1), which in my opinion didn't work at all and was a prime example of Rodgers' defensive naïveté. By contrast, Klopp only saw Allen as suitable to category #3 - so in those rare games Allen was playing, he was operating much closer to the opposition goal than at any time under Rodgers. Klopp values athleticism in his midfielders, so using Allen as a holding midfielder would be a non-starter for him while as a playmaker Allen is too reliant on short passing for Klopp's liking. This shows that the categories above can be very subjective, even for the professionals.
A couple more things on the attacking CM role. This position exists only in 4-3-3 type systems; in 4-2-3-1 it will probably be midfielders of the first 2 types playing together (i.e. Can+Henderson). However an attacking CM can be used in 4-2-3-1 in the #10 role - that's what Pochettino often does with Dembele at Spurs or we can just remember Gerrard in his prime. Stevie left such a void, that we never really replaced him with another attacking CM, which in time made our attack less purposeful and direct and more prone to pointless tip-toeing around the box. I think this is why Klopp has apparently considered an attacking CM to be his priority purchase this summer.