This is why even as a hardline atheist I have sent at times my kid to private religious education, and I hope I can continue to afford it. Neither system is great but worshipping someone who exists only as an ideal entity in your mind seems marginally better than worshipping the world's largest incarceration machine and military aggressor.
I had the opposite experience, I never did the pledge in public school (or in the Adventist schools), at least not regularly enough for me to recall, but in the evangelical school it was the entire school doing it together in a big room every damn morning, first thing. I'd argue religious schools are more likely to drill it into you than public schools on account of the "under God" part that they've retconned in.
I also never saw military recruiters in public school, neither in military-dense Hawaii nor in deep red central Oregon.
There was never an assembly dedicated to them; it was always just a kiosk-type thing students could walk up to.
Did you hear a "devil's advocate" argument for the burning of Tulsa, Oklahoma by white supremacists?
Both-Sides-Ism isn't a real way to account for ideological coercion in education. It's just painting a picture that includes your enemies. What your enemies look like in the picture is LADEN with ideology.
In religious schools, even the best of them, I learned nothing about any ideology but the “true” one, everything else was people being deluded by the devil.