I don’t carry my green card with me except when I’m travelling internationally. Technically I should, but it can take up to a year to replace if lost and I don’t want to lose it.
I’ve never once been asked for it in 30 years (except when travelling internationally of course). I suspect that might have been different if i were not a white person with an “i love your accent” accent! My drivers license does not in any way identify that I am or am not a citizen.
I’m pretty sure some US citizens in the small city I live in have never possessed a passport. Our neighbor had never left the state when we first met.