Frankly, I think the argument to support Kennedy was pretty compelling. The first amendment allows for the free practice of any religion by the people. It makes no commentary on the government specifically censoring any religion. In fact it appears to be just the opposite. The text reads that the government has a duty to not act at all when religious expression is made by the people.
And that's all in public schools!
From Sotomayor's dissent, page 18: https://www.supremecourt.gov/opinions/21pdf/21-418_i425.pdf
It's important to note that coercion doesn't have to be explicit with observed punishments and rewards, and Sotomayor explains this quite clearly (including how courts have previously held this to be true for many years).
Like most SCOTUS opinions and dissents, these are technically well-written, accessible, and easy to read - I would encourage you to do so. They sometimes even border on enjoyable to read (although maybe I'm just weird). The case as laid out in the dissent is rather egregious, and it's shocking that the SCOTUS upheld such a clear violation of the establishment clause. It's shocking that they even heard the case to begin with.
The school district practically begged the coach to work with them to find an avenue for him to express his religious beliefs through prayer, without creating a coercive environment for students. At every opportunity, the coach declined to find a way forward that didn't violate the establishment clause.
It's important to know that "prayer in schools" is a rallying point for conservative christians, and the coach's behavior was intentionally provocative. Holding big, public prayers like this is a strategy designed to continually test the limits of the establishment clause, with the goal of ultimately breaking it.
A piece of anecdotal evidence that I always think about when stuff like this comes up - I went to a very, very conservative christian school. Prayer was pervasive (as you might expect), but it was never used to make a public scene like this. Ever.
Finally, his behavior is an affront to prayer itself, which Jesus himself makes clear in Matthew 6:5-6.
> And when you pray, do not be like the hypocrites, for they love to pray standing in the synagogues and on the street corners to be seen by others. Truly I tell you, they have received their reward in full. But when you pray, go into your room, close the door and pray to your Father, who is unseen. Then your Father, who sees what is done in secret, will reward you.
It says a lot about his motives. That's not exactly an obscure passage of the Bible.
In this case, I would say yes, the coach should not be allowed to pray because the prayer was done as part of his role as a coach. Like if a social worker prays that a child is ok while driving to a home visit, then that’s ok. But if the social worker forces the family to pray then that’s different.
Same logic applies elsewhere. Should governmental employees be banned from wearing hijabs(they’re a public form of religious expression)? Should we ban/fire a Muslim employee for praying in public(if there’s no available private space) while working as a member of TSA? Limited powers on governmental religious discrimination is a good thing.