Here in the UK we have procedures to meet all four of your requirements, and checking boarding passes at the aircraft door contributes to meeting number 4.
The boarding pass is issued at check in, which is the first part of the screening process. It is then checked again at airport security to ensure only checked in passengers can gain access airside. It is checked at the gate to reconcile boarded passengers and to ensure that passengers who have checked in online have their identity verified and passports checked for validity and visas (airlines get fined if passengers travel with incorrect documentation). Inspection at the aircraft door is a final check to help catch any mistakes that may have been made earlier in the chain. Airport staff are only human, they work long, unsociable hours, and occasionally a passenger may slip through the net and try to board the wrong aircraft.
Making sure that all passengers have a valid boarding pass when they board the aircraft actually helps to makes sure that "no people who haven't been screened are able to board a flight"