Coaches from the McVay tree will be drooling over Washington. They all want that big Tyler Higbee, Kittle, Marcedes Lewis type of TE who can block like a 6th lineman, so they can run the ball while still lining up in 3 WR sets. Washington can do that right away, while maybe developing as a pass catcher later. He could be a key cog in that offense.
I don't actually know that much about the McCarthy / Schottenheimer run scheme, but I don't think TE is a key piece in it. I can't even remember a TE who MM had in Green Bay, aside from Jermichael Finley, and he was basically a giant WR. Good player too.
Anyway, if TE's not a premium position for McCarthy, I would be fine with rolling out Schultz or Ferguson and spending our 1st rounder on a player with higher impact.