You make reference to the constraint, "that met certain parameters."
There needs to be a mechanism in place to enforce such a constraint. My point was, sysadmins are in a position to skate right past that mechanism (part of their job is maintaining the facilities that enforce the constraint for normal employees). The only real check on sysadmins is other sysadmins.
Who knows what resources Snowden had access to? E.g., perhaps he could have recorded POTUS's conversations on a USB stick. But for how long without being noticed? And for what purpose? And for what claim to fame compared to the one he now makes?
If I were he, I'd have loaded my USB stick with the most damaging possible rosters of NSA information. Then, I'd have made copies, placed them in envelopes addressed to suitable organizations, and entrusted them to different folks with the admonition, if anything happens to me ....