Which countries? I fly a lot NL<>ES, ES<>DE and they don't check my passport usually. I do take it because sometimes they ask for it, but mostly they don't.
The UK obviously being the outlier; very paranoid with extremely old scanners so security takes a long time, especially at Heathrow.
I don't believe they run any inter-schengen or other "domestic" flights, and the 5th freedom flights they do run are like Singapore to Sydney so attract passport viewing.
Security is irelevent to BA's problems.
Airlines like Ryanair do check photo ID on domestic, but that's for revenue purposes (to stop people selling tickets on the black market)
The Police do have far-reaching powers under the fascist "Schedule 7" of the terrorism act 2000 to demand ID of some sort in certain circumstances, but that's not unique to flying.
For EU flights especially you still check passports at the gate - indeed it's the only place they are checked.
For other flights they normally check visas at the checkin desk, but if you're transferring you usually get visa checked at the gate. Most people have nice simple visas that pretty much anyone can check, a few troublemakers like me rely on more complex timatic rules (say TWOV in Beijing on multiple tickets), but no reason that can't be checked at gate either.