Unfortunately I don't have any quality headphones.
I listened again with better speakers though. I suppose you are correct... what I thought was messy pedaling does seem to be the added reverb. In any case, I think I would prefer without the added reverb, or with slightly less. Either you make a recording in an acoustic environment that has great reverb, or you must live with a slightly drier-sounding recording. Did you record this at home?
There was some lovely rubato throughout and fine attention to the tenor voice. My quibbles with missing inner notes, upon second listening, seem to be minor.