No, you fail to understand a light cones end at finite points. If I travel 1 light year, then I observe a different light cone and it takes 1 year for somone that stayed behind to receive information from me. This effectively means you can see something before the light cone at your current location would observe it.

If you consider a light cone as all points out to infinity then that’s the entire universe. Instead at each point in time you have a different light cone that extends to the age of the universe.

In theory we can never see past a second limit based on the expansion of the universe, but that’s yet to be observed speculation without any consistent theory of what’s going on.