They are right, but there is a behind-the-scenes part that nobody bothers explaining and complicates things for you.

Thing is, at some point before departure, typically T-72 hours, flights go to "airport control". At this point, reservations can no longer touch seating and possibly other ancillaries.

At this point, flight becomes possible to check into, as you did via UA. That does not mean that the flight is available to check into on OLCI pages of Swiss, as that is dependent on their access rules (often 36 or 24 hours before your first leg with the specific airline, I don't remember which number Swiss goes with). Either the fact that you did OLCI via UA (UA system forwarding that to LX system) or possibly just the fact that the flight went into airport control triggered seat assignment for you.

The agent was right both that they can't change anything anymore (their access to the seating reservations are blocked), and that you will have to wait until Swiss opens OLCI access to you directly.