We know how exposure works but Klopp can be quoted as far back as to 2018 speaking against these kick offs. So considering your continuous conjecture, Klopp has been regularly and knowingly been shafted by the club every single time this has happened in the past 5 years, which is around double digits. So it's even more confusing when you say "So it will be up to Klopp to speak out again."
Firstly it isn't the clubs choice when it plays. Each weekend the TV companies get to pick which games are televised.
BT/TNT bought the package for thirty two 12:30 kick offs per season. 20 of those fixtures are 2nd picks and 12 are 5th picks.
So the only way we get a 12:30 kick off is if Sky do not pick us.Then you get the situation in which certain games can only be played in certain slots. For instance Europa League teams can only play Sunday after a Thursday game and the Police get a say on high risk games.
Currently the maximum number of 12:30 League games a club can play is 6 per season. Liverpool could easily lobby for that to be reduced. The other clubs who don't get as many games would be delighted. The issue for LFC would be that we would be cutting our income whilst increasing other clubs income.
Klopps job is to look after the players welfare and to win as many games as possible. The commercial side of the clubs job is to bring in as much revenue as possible.
Klopp has made it clear he doesn't like pre season tours. That hasn't stopped us going to the Far East for the last couple of seasons and lining up a trip to the States for this summer.
The brutal reality is that if you want to compete at the top then getting as much revenue from TV and tours comes with the territory.
We are contracted to play up to 6 12:30s a season. That can be changed in the next TV deal currently being negotiated and the club could push for fewer games but I wouldn't hold my breath