I always value what is shared from experience over what is merely learned from a book (or whatever). Thank you.
You made an observation:

Many people believe that you should allow a child to choose their own path.

and this is certainly true for many these days. I would observe that children tend to choose their own paths regardless of what we think we "allow"! All we parents can hope to do is equip them to make wise choices. And it is our daily behavior that equips them, as you point out.

The authoritarian, sheltered strategy typically fails the first six months the child is away from the authority figure, i.e. shortly after they leave home. The other extreme of complete liberality simply fails sooner, usually between the ages of 12-14.

Who we are is inculcated into our children, for better or worse. If we we love our children become loving. If we are vengeful our children will be vengeful. So yes, "your values and beliefs are the foundation" of who we are and is manifested in our behavior. It is our daily behavior that shapes our children. Children become what we do, rarely what we say.
Observing our children as they reveal us to ourselves can provoke the wildest of extreme emotions. Everyone should have a few ;-)