No, it shouldn’t. Haven’t tried on my own but I am very certain this is how Signal is designed: Messages you’ve received on your phone won’t be available on your desktop app unless you have an internet connection on your phone. So linking a device with your account somehow (I’d have to research as well ) exchanges necessary decryption keys to read the messages. But since no content is stored on the servers, any messages not received directly on the linked device currently in use need to be downloaded from the device they originally were received on - be it your phone or yet another desktop client.
There’s been an article on a German tech site comparing sync behavior of Signal and Telegram (which I can’t find atm).
They noticed that for the case of Signal taking the phone into flight mode, no old messages appeared on a freshly linked desktop client.
While for Telegram all messages were easily accessible to the desktop despite having the phone disconnected from the net.