Private schools can do what they want so long as there is clear communication with students and families and what they ask students to do isn’t illegal. You can always decide not to attend that school.
Public schools are paid for with tax dollars and thus answer to the public, via the school board, town, department of education. If the majority of the tax paying public in your school district are okay with the pledge of allegiance and such, then it happens. Whether these actions are explicitly political is debatable, but I personally don’t find a moment of silence political.
In end, your student can abstain. My daughter goes to a Catholic School, but doesn’t participate in the morning, school-wide prayer. She just stands there with her eyes closed and collects her thoughts for the day.
There are no nuns beating her with rulers in this day and age.