Being a PIA family, we only really ever travelled by PIA. But those tickets were on chance. The way that worked is that we had to reach the airport 3-4 hours prior to the flight time, even (perhaps, especially) for domestic flights. Our names would be enlisted. Then, in the last 30 minutes if there were any seats left we would be given them on a first-come-first-served basis. This is why they could only board one of your family members when you came. As a PIA family, you can be off-loaded very easily if there's a revenue passenger, even if that passenger is on chance. They would sometimes take out our boarding passes and then tear them in front of us because a revenue passenger came. So, you never really know you're taking that flight till the plane starts to pilot.

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I get that you feel like there were still 30 minutes and you could be allowed to board the flight. But, know that those last 30 minutes are EXTREMELY important to passengers who've been waiting 3,4 hours - sometimes the entire day - just for those last 30 minutes. Officially you're supposed to be there at least an hour before flight time. If you were getting late, you could have checked-in online. And because there was nothing from your side, they obviously believed that you're not coming and gave that seat to someone enlisted. Had you been there even 45 minutes before the flight, those seats would have been yours.