Scott Chandler who played TE for Iowa is 6-7. Besides Spaeth and Chandler, other notable tall (> 6-5) TEs in the pros include: John Carlson (6-5), Jermichael Finley (6-5), Kellen Davis (6-7), Jermaine Gresham (6-5), Todd Heap (6-5), Evan Moore (6-6), Brandon Pettigrew (6-5), Heath Miller (6-5), Kyle Rudolph (6-6), Jeremy Shockey (6-5), and Jason Witten (6-6).
I'm not saying Johnson belongs in the same category as the guys I've listed above, but it's not like his height disqualifies him from the position. From what I've seen of him on film, he looks to have some quicks, but that's against fellow high-schoolers. He's the kind of guy who probably should stay at TE until he plays himself off it and hopefully, he never would.
TE has become a real hybrid position at both the college and pro levels. They come in all shapes and sizes and their use depends almost wholly on formation and situation.