There is a limited number of pass holder seats on the Eurostar and on popular departures and during high season those sell out weeks, and sometimes months in advance.
The best place to see the availability of passholder seats on the Eurostar is
https://www.b-europe.com/EN/Booking/Pass#TravelWish
Press "later trains" and eventually you will find the next available connection.
If you have a mobile pass you need to generate a Pass Cover Number in order to make the reservation at b-europe. You do that here in the PCN generator:
https://community.eurail.com/news-and-announcements-39/pass-cover-number-generator-is-live-5653
To access the train you have to pass a gate that needs the QR code on the reservation. They then separately check your pass QR code before boarding.
To access the train you have to pass a gate that needs the QR code on the reservation. They then separately check your pass QR code before boarding.
Have Eurostar actually checked your pass? I've read multiple reports of frequent Eurostar users (with pass) that they have only ever had to show their reservations...