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).
I'm sure it will become possible soon. The code is already there on iOS, as the app also work on iPad, but hidden behind the internal feature flag [0]. Same with Android [1]. If your second device in an Android, you can already use it now with [Molly](https://github.com/mollyim/mollyim-android).
Also, WhatsApp recently added this feature, so the expectations from potential new users who switched is now there.
From my understanding, they're not a fan of it (not sure if it's officially against their TOS or not) but they don't go out of their way to stop them. At least as long as you don't use the Signal name and make it clear you're not an official app.
Even in this blog post about usernames, they clearly make sure to mention them: "This means that in about 90 days, your phone number privacy settings will be honored by everyone using an official Signal app."
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).