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I'd first try removing the auxiliary drive belt and turning every pulley and ancillary ( power steering pump , alternator etc) by hand. All should turn easily (with the exception of the AC pump) with no grittyness or wobbly pulleys- If you do find a sticky pump / alternator then it should be replaced or repaired
If the engine is stone cold you can briefly run it with no belts to see if the noise changes or has gone away, this will help show if an ancillary is the issue
Another possibility is the variable valve timing system as this does rely on oil pressure to correctly synchronise your vvti pulley on the end of the camshaft. If the valve has become blocked up then its possible the pulley is rattling about and producing the noise.
if you've got a top-end rattle or regular ticking that increases in tempo with increasing rpm then yes I'd agree it could be a sticking valve lifter, as these use a small hydraulic piston to take up the valve clearances and what can happen is the small oil ways that feed the piston chamber can get clogged up
If you haven't already then I'd try an oil flush treatment complete with a new filter and fresh oil
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