1. You receive a message from someone.
(wait a few hours)
2. You open this tab/read the message
3. The number of unread messages counter doesn't update.
The linux client... well, for the longest time you could be lucky if it would even work. After years of not updating the bundled libraries, they stopped interfacing well with the ones in the system.
And then there was an update, which changed nothing, except for introducing ads on linux and disabling video chat for most cases.
The issue of skype might not be P2P, it sounds more like it was a lack of ability to provide a stable product of any kind. The Android app was also the worst app I had seen for many years. Who the hell seriously builds their own UI toolkit, and then makes it work worse than the one bundled with the OS?
Funny thing the company DNA, P2P is still haunting Skype.