Reciting the Pledge of Allegiance at school, or singing national anthems at non-national events come to mind as rituals not affiliated with the supernatural[1]

I hazard a majority of Americans (religious or not) voluntarily partake in those rituals.

1. I suppose rituals require venerated objects/subjects, and this is often religious, but does not have to be

This is addendum, not correction.

Rituals have a few distinct elements:

+ Repetition both in the ritual and of the ritual itself.

+ Assigned time or place. Eg praying before dinner, or Christmas mass.

+ Fetish. Some kind of object or words, a "solidification of intent". This can be a garment or some specific movement too.

So yes, an anthem is absolutely a ritual, as is a specific cheer for a sports team, under this framework.

Rituals serve several purposes including fostering group identity (the family praying before dinner together), marking something as special (the anthem before a sports game) and demarcation (Christmas mass). Obviously a ritual can fulfill some or a blend of these objectives.

Rituals, I believe, matter for the subconcious and non-rational parts of the human psyche. I think intentional, religious or non-religious, rituals are a useful tool.

Sort of, the pledge includes a “One nation under God” bit but few take it in a religious context.

That also came much later and if we take it in the same sense many of the founding fathers used the word "God", it would not be compatible with mainstream Christian theology.