For what it’s worth, this issue has been reported for a long time across different distros, so it’s most likely something on Valve’s side. Maybe some kind of timing issue or perhaps it depends on the type of update being processed.
For me, usually the first boot-up of Steam on a fresh install it does what you’ve described above. Subsequent client updates will pop-up a little progress bar modal to indicate it’s updating… Most of the time. I’m quite sure I’ve caught it updating without a modal popping up on existing installs as well.