Spurs boss Andre Villas-Boas has been criticised for allowing goalkeeper Hugo Lloris to remain on the pitch after a sickening clash with Everton striker Romelu Lukaku during Sunday's 0-0 Barclays Premier League draw at Goodison Park.
Poyet had withdrawn his own goalkeeper Keiren Westwood 24 hours earlier after he also took a blow to the head in a collision with Hull defender Paul McShane, and the 29-year-old Republic of Ireland international remains a doubt for the Black Cats' Capital One Cup fourth-round tie against Southampton on Wednesday evening.
However, the Uruguayan is convinced the final say should be left with those who know best.
He said: "I think there have been too many people talking about this.
"My opinion is that we should respect the medical staff a little bit more and let the ones who know about it talk, and not the rest who don't have a clue about it.
"There are too many people talking and it shouldn't be like that. It should be the medical staff really, the ones who know and the ones who jump on to the pitch and need to measure a concussion in seconds.
"Now if anyone can do it, you are more than welcome to football. I can't do it, but the rest all have an opinion.
"You need to know the person, the player, if they are willing to play - plenty of things come into it and it's not that easy."
Westwood was left dazed by his brush with his international team-mate, and Poyet revealed he has only a 50-50 chance of being involved against the Saints.
He said: "It was the back of the head, the neck. He was feeling quite dizzy. It wasn't a great weekend for goalkeepers in the Premier League."
Poyet will be forced to make at least two changes for a game in which the winners are assured of a home quarter-final clash with his former club Chelsea, with midfielder Lee Cattermole and full-back Andrea Dossena having both been sent off in a calamitous first half at the KC Stadium.
It was the seventh red card of Cattermole's Premier League career and while Poyet felt it was harsh - Sunderland have decided not to launch an appeal against either decision - the manager admits he will have to succeed where his predecessors have previously failed if he is to rid the former Middlesbrough and Wigan enforcer of his reputation as a rash tackler.
He said: "In any team, you have got different characters and you need to, as a manager, use them to the best of their ability knowing they can do one thing or another, or they can make this mistake or another.
"Lee, we know that when he is frustrated he is maybe going to make this kind of tackle.
"Many managers have talked about him. Me, I am going to try to manage him my way.
"I respect other opinions but I am going to do it my way with him internally, and we will see if he improves that reputation that he has got at the moment in England that he cannot make a tackle."
A 1-0 defeat in East Yorkshire took some of the gloss off the previous weekend's derby victory over Newcastle, the Black Cats' first in the Premier League this season, and left them five points adrift of safety.
However, while top-flight survival is the overwhelming priority, Poyet is relishing the encounter with Southampton, English football's latest surprise package.
He said: "They have done very well. They have kept the core of the team. They have added two or three players who are performing at great levels, so they are a very strong team.
"They are the biggest surprise so far, but it's totally deserved because they play a football which is quite brave and very aggressive, especially pressing at the beginning of the game, so there are plenty of things we need to do right to win this tie."