The Desktop app is definitely independent from your primary device, once it's been linked. The WhatsApp desktop app used to require a connection to your phone, but even they updated it recently to the same architecture as Signal, where each device connects directly to the server.
If you don't open the Desktop app for a few weeks though, there is a "syncing" step where it fetches the recent messages queue from the server (can't remember the exact number, might be the last 1000 messages or all messages from the last 30 days or something similar).
If you don't open the Desktop app for a few weeks though, there is a "syncing" step where it fetches the recent messages queue from the server (can't remember the exact number, might be the last 1000 messages or all messages from the last 30 days or something similar).