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Tailscale uses the same range as CGNAT.

You shouldn't be seeing the CGNAT addresses inside your home network, should you?

Depends on your network, router configuration. But in any case, if you install it on a router where the upstream uses CGNAT yes, you would.

Router straddles inside and outside. I know there's a way where you could put tailscale on your router but it's not the normal way to use it.