Why can the police only take action against an alleged offender if a report is made?

Without a report, there is no complaint.

No complaint, nothing to act on unless someone is caught by an officer in the act.

How else could this work? Thought crime?

If an offence isn't reported, how will the police know if an offence has been committed?

they do not need a complaint to act on

if they witness a criminal event they can and probably will make an arrest

however the majority of the time the Police do not witness Criminal events and other citizens do
in those cases until some witness reports the event the Police have no knowledge of it and therefore cannot take action

much like you being unable to Swat a mosquito when you don't know it is in the room