That doesn't change the fact that Arteta had years of positional training that Khedira did not have. Those skills were essential in helping him adapt to that deeper role.
I understand the desire for more physicality in midfield (and I'd like more too), but I also think that a lot of general weaknesses in the team appear most obviously in the central midfield area. There's more going on than just Arteta being old and slow. Against Liverpool and Chelsea it was our inability to keep possession under pressure (which was freely applied because of a lack of pace and cutting edge in behind), along with super-high fullbacks, that set up wide-open counterattacks at pace. The Liverpool 3rd/4th goals and the Chelsea 1st/2nd/3rd/5th all came from that exact situation. Similarly, the 3rd against Everton came from Sagna losing the ball out wide in midfield.
Why haven't Mourinho Chelsea ever lost big? They keep numbers behind the ball. Why didn't prime Barcelona ever lose big? They could always keep possession under pressure. How does Theo help us simply by being present on the field? He pushes defensive lines back and creates space for our midfielders. I want a new CM, but I don't think it's smart to make our big game problems about the 'DM' first and foremost. If we're going to play high fullbacks (and therefore not keep numbers behind the ball), we need to keep possession under pressure and pose a threat in behind to keep the defense honest. Our biggest problem IMO is giving away open counterattacks in big games. Khedira by himself doesn't change that. In fact, his less than stellar qualities as a passer could make it worse.