I have been inspecting all three log files to see if I could find any useful information. In an attempt which I successfully launched Fusion 360, I tracked them to see the updates.
Inspecting the log files in real time, I found something unusual in the app log file. Every minute it constantly writes a line:
20211223T085005 I 3792 PUT 200 https://f360sc-p-ue1.api.cosv2-p-ue1-dn.autodesk.com/api/v1/sessions/59410421 - workflow:SLM_heartbeat attempt:0 requestId:f360_SLM_heartbeat__1-61c3c7bc-23f7446848e94d6798df34db dt=0.69
which I absolutely did not understand. Naturally, to understand it better, I copied and navigated to the url in the log message (https://f360sc-p-ue1.api.cosv2-p-ue1-dn.autodesk.com/api/v1/sessions/59410421 - workflow:SLM_heartbeat attempt:0 requestId:f360_SLM_heartbeat__1-61c3c927-df434d568d8d46379812c21a)
It showed a Missing Authentication Token message:
While it still does not mean anything to me, I am expecting that it might help to diagnose the problem. All three log files are attached with this reply, in case anyone is interested.
Besides that, I tried to reproduce the freezing problem by calling:
app = adsk.core.Application.get()
app.userName
in the text command window. (I believed that this action triggered the whole freeze-at-launch thing). The time of this action was around 0849a.m. on 23rd December 2021 (20211223T0849xx in app log file time format).
However, there was nothing special in the log file (specifically, app log file). It still updates the same message (Missing Authentication Token) every minute, with occasional Runtime Error messages.