What you really need is a good old-fashioned Flight Engineer to put you pilot-type chaps (not all, but many) at rest with your perceptions of air conditioning and pressurisation theory.
Ram Air was mentioned... and that's where the initial mistake was made. Ram Air is used within the primary stages of 'Heat Exchange' within the Air Cycle Machines and nothing to do with air entering the cabin.
BOAC appears to be the only chap here to have a grasp of the situation. Over the years, the method in which the pressurisation of aircraft is achieved, hasn't changed that much. Yeah, more modern systems have 'tweaked' the theory in order to refine passenger comfort.
A modern system will start to close the outflow valve(s) during the takeoff roll, or, whenever the Pressure Controller has been selected to AUTO. This is in order that there will be a much smoother transition to pressurised flight when those tricky little squat switches handover full control the the Pressure Controller. The Pressure Controller will sense how many packs are ON/OFF, and, it will also know the position of the Outflow Valve(s) and that it modulate the Outflow Valve(s) with respect to the rate of change of cabin pressure.
Simples...
TCF