UE4 Documentation strikes again …
From what I can determine from the timings of the Log messages is - When the code is executed on the server there are a lot of other stuff going on between the constructor() and init() but when executed on the clientside the Init() comes immediately after the constructor().
I am using UE 4.26.2. I never got the Init() Log message not being displayed. I always had both log messages displayed in the output temp logfile. Either as running in PIE as standalone or as a multiplayer session.