It could be worn bores or worn or stuck or broken piston rings, causing crankcase pressure. Try removing the oil filler cap while the engine is running, and holding a piece of paper flat over the opening. If the ventilation system is drawing properly it should be sucked gently in. If it blows out then there is either excess crankcase pressure, or the suction in the crankcase ventilation system isn't sucking enough.
That could be the small pipe from the flame trap to the inlet manifold, and the small orifice in the nipple, being blocked.
When the engine idles or there is high vacuum in the inlet, the small pipe draws air into the manifold from the flame trap. When the throttle is opened the vacuum drops, but there is then suction on the air intake side of the throttle body, and the large pipe to the flame trap draws in air, again through the flame trap.

So either way, if oil is getting up into the flame trap it then gets sucked into the inlet.
The oil separator black box has to be free from clogged up residues. An airway through isn't really good enough, because then it doesn't catch the oil properly, which then passes on up the pipe to the flame trap.
I'd do a compression test, as suggested, then thoroughly clean all the breather system. preferably fit a new separator box.
Could be failed inlet valve stem oil seals, or a combination of all these things if the engine has done a high mileage. The usual way to distinguish oil burning from worn bores from worn inlets is to note the circumstances when it happens.
Blue smoke on acceleration = bores, blue smoke after idling or deceleration, when you open up again = valve stems.

There have been threads on replacing valve stem seals without removing the head - see my own contribution from some months ago.

I hope some of this helps.