Comment by Timon3

No, you did not. You specified that you're only looking at competition, and then replied to yourself that there isn't anything else.

In case you really can't imagine alternatives: what if the private system spreads lies about the non-private system? What if they use their influence to change laws and regulations to prefer their system? What if they use their influence to make others with influence treat those using the non-private system worse?

None of these are competition in the sense you're asking for, and yet they are a natural consequence of private incentives.