He can definitely move. That's not the issue I have with him. But with most TEs, they aren't going to just line up and beat the guy in man coverage. The way LBs move now and then safeties, I don't think there are many TEs that have that twitch to separate at the top of their routes. We see it with Kmet. When he is open, he is schemed open. Misdirection, playaction, rollouts, chip block before releasing, screens. But he's not going to beat man coverage. You're not going to line him up in the Y or the slot and him separate from his defender. It's not a knock on Washington or Kmet really, it's just that 260 and 270lb men can't do that well in general. Kelce, Kittle, Waller, Engram are 240-250ish and while Washington may be able to run in a straight line with any of them, I don't see the quickness in and out of breaks with him to make him a weapon like they are in the straight drop back passing game.
120 catches from TEs is about 180 targets at the position. That's a lot for the Shanahan offensive scheme. Also a lot for a team that doesn't have an elite level TE. Like Washington and Kmet at 5-10-15 yards at a time, as you said, isn't enough of a big chunk to warrant that many targets. Obviously, there's still plenty of targets there to take your deep shots, but you're also not getting a bunch of YAC on those throws either, like you would with a WR outside or a RB out of the backfield, and that takes away from those targets. Again, not that Washington isn't great after the catch, but he's also likely to be making those catches in the middle of the field where the bulk of the defense is there to tackle him, as opposed to WRs/RBs on the outside facing 1-3 defenders with the capability of getting outside the numbers to make a tackle.
I like Washington a lot. I would be happy if they drafted him. But 2nd round is too rich for my blood. Even that late Ravens 2 is too high. I probably wouldn't feel comfortable taking any non elite level TE in the top 75. It's why the Kmet pick was always "meh" to me. Good player, but just doesn't have the value to match where he was picked. Washington would be in the same boat, IMO.