Yes, it is a violation of the First Amendment for the government to suppress other people’s speech. But it’s not suppressing other people’s speech, it’s suppressing its own.
The government can choose what it wants to say. It can’t choose what it wants other people to say.
You’re assuming everyone who participates in this sort of training is a government employee when they might not be. And given the size of the federal government, it has an impact on the speech of other institutions.
They are making a political decision here to exclude things that make the current administration angry. It’s a political decision, not one based on what is best for this kind of training.
It is a violation of the first amendment. The are both forcing federal agencies to stop paying for any training that involves critical race theory but also monitoring employee activities, including internal employee discussions, and attempting to ban teaching of thr 1619 cirriculum in public schools.
So they are restricting the speech of fed agencies because of the reason for prohibting these trainings. There is an argument to be made, though not a good one, that this might not be a 1st amendment violation. But they are also restricting the speech of individual federal employees and public schools. Both are definitely violations of the 1st amendment.
I’m talking employee bookclubs and internal meetings. https://www.google.com/amp/s/www.politico.com/amp/news/2020/09/09/education-department-race-related-content-411487
It’s well settled black letter law that the government can choose what it wants to say. This includes what it’s employees say while on the clock. To say otherwise would be to say that the government must allow their employees to proselytize their particular version of Christianity while at work.
The government can’t choose what the employees say off the clock, and if they want to have unofficial meetings to discuss this sort of thing, they can.