Speaking as an extrovert who does public speaking as part of my working life - I don't think it's gruelling at all. I would imagine it's very taxing for people who are introverted who would prepare what they're going to say and then spend ages analysing their conversations afterwards. In a game where psychology and mindset is such an important part of the game - I would imagine there would be a noticable advantage to extroverts over introverts. The two ways of solving the problem are to ban the press, which would haemorrage money from the game but be fair - or insist that all players do press time relative to their ranking whilst mitigating as best they can disuption to the game.

A simpler case is; any time spent with press is not time spent training, doing match analysis or concentrating on your mindset for an upcoming game. This would be especially true if you were doing press after losing a match.