I'm not sure exactly what you're describing, could you provide a save or image (with path reservations on)?

I wish stations had built in signals (2) but it would be a major change. Currently you can usually get the same behavior by putting two way path signals on the end(s) of your station.

At this station with 2 tracks traffic usually flows fine. If a station track is occupied, the next train takes the other. Problem is only when a train has left a station track but not yet the tile after with the signal, and breaks down there.

I personally have run into situations like this where trains get stuck deciding at PBS. I don't know if the devs would consider it a bug though. It may be possible to fix/improve based on some yapf settings.

For this situation specifically you can fix it by using block signals at the entrance instead of PBS

the problem here is that pathfinding and path reserving happens in two separate phases:

1. the pathfinding adds up penalties until the destination station is reached, and then picks the platform with the lowest penalty
2. the path reserving reserves up to that destination platform, and then a little bit further until the next safe waiting point (signal or end of track), however, this additional bit of track was not part of the pathfinder penalties earlier, so there were no penalties added for if these tiles are blocked

This bug was also reported, with another example savegame and explanation, as #11506.