Elliot started a lot last season and struggled though.
There were a lot of factors at play; the rest of the midfield was poor, the system was different and he has improved his play since then, but he also struggled a lot when teams would throw everything at us in the first half. He was often brushed aside quite easily, teams would cut off the left footed passing option and then close him down quickly, and he'd often play casual passes into incredibly dangerous areas (that dummy against Brentford still baffles me).
Using him sparingly this season has been an excellent choice both for him and the team, he gets to come on when players don't have the energy to close him down quickly or physically bully him, and as a result he gets to impose his game on them. He's had some really good games when starting too, but he often starts in games that are very open or against slower European opposition, both of which suit him well.
I think the above is why he doesn't start more, with us instead leaning on much more physical players like Dom, Jones and Gravenberch. The question is whether his game has developed enough to the point where he can start against those more aggressive teams now, I'm not sure of the answer but if Klopp isn't starting him in those games then my assumption would be no.
There's also the question of whether we actually need to start him, the rest of the midfield is pretty strong, he's decisive from the bench and he's not exactly angling for a move elsewhere.