He doesn't have to request a trade. If he is traded, he probably won't - it will be a game of chicken/optics.
Let's say we finish in the basement again this season, look at the situation you're in if you're Staios and Brady says anything but "I want to stick it out here for the rest of my career, I knew we just need a few more years of re-tooling and we can be a contender, I love the Carp fair and they don't have a comparable event in other cities."
Then Staios is in a situation where he knows he has a demoralized Brady, his NMC is about to activate, and even if they keep him they have to explore trading him in 1 or so seasons anyways because it isn't like he has lots of term remaining. He has 3 years left starting in 2025-26. Teams that are not winning typically try to engage their stars in their willingness to stay when the star has 2 years left, because that gives them more runway to make a great deal (see Chychrun).
Ultimately, I cannot fathom the team keeping him after July 1st 2025 if we miss the playoffs in 24-25 and Brady doesn't give a strong and firm indication that he trusts the process and wants to be here for life. He won't ask for a trade, and ultimately that is better for both sides. It is better for the player optics wise, and it's better for the team in terms of negotiations with other teams. (With that said, other teams aren't stupid and can recognize the choke point that comes with July 1st 2025).
The Matthews situation in Toronto was somewhat similar, except easier for them to manage because he was eligible for his extension. They engaged him prior to July 1st, and he made them feel comfortable that he wanted to stay in spite of all the first round exit stuff, so they took the risk of carrying him beyond July 1st because it was reasonable to expect that he would stay. Had he said "I'm not sure I want to play out the season", maybe they would have had to try and get something for him. Similar to if Brady says that he isn't sure about the direction of the team and doesn't trust we are close enough for him, then entering into a situation where his NMC activated and they probably have to trade him anyways a year or two later from a massive position of weakness would be strategically incompetent.
I'm not saying he's gone, but this is all common sense following how the league works in terms of player movement, NMCs, how star players operate when they have the leverage to get out of a bottom feeding team and go to a contender, etc.