Lack of compression will keep it from starting/running.
Smoke from the tailpipe or from under the hood (burn-off of the oil splashing from the missing cap)? If someone failed to replace the cap after adding oil then maybe, just maybe, you could be in the clear. But it sure sounds like there is some massive overpressure in the crankcase caused by blowby in the cylinders (worn rings). Usually the PCV box spews oil all around the oil filter housing, not the fill cap though.
Look for anything else gross and obvious and check the oil and coolant levels, top them off then try again. Not much to lose at this point.
Best of luck,
Will
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XC60 / Odyssey
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