He has all those accounts blocked. What I suspect Pat does is he perforns search for accounts that piss him off. Either than or he has the all saved as bookmarks on his phone. He has to go to each account individually. He then clicks on the button so he can see that specific accounts tweets. He then manually scrolls through each account looking for tweets to respond to.
He can't just search his own name because blocked accounts won't show up in the resultant twitter search.
And since this takes so much time, he doesn't bother to untag random accounts his trolls add. Hence why he ends up telling president, pope, the CIA, his local PD, FBI, ect to "enjoy prison".