I’m going to say it is not your engine. It is probably your belt, and belt tensioner that needs to be replaced. And it’s probably a good idea to change your water pump too. if it has never been changed.
Concur w/@weekend-warrior above, noise description seems unlikely to be caused by lack of oil on internal engine parts. To confirm, ask your shop to measure the engine’s oil pressure using their shop gauge. If that’s ok, ask if using a mechanic’s stethoscope might be helpful to find the actual source. Adding additives to the oil could have an adverse & expensive to repair unwanted effect, particularly on the variable valve timing components. If you want to do something w/the oil, an oil and filter change can only help, provided the job is done correctly and the correct spec’d oil and filter are used.
I think you need a better mechanic.
After ONLY 13 years? 
Yup!
+1
The mechanic is probably wrong. A better mechanic or DIYer would probably remove the serpentine belt and briefly run the engine. The noise will most likely be absent - most likely it was caused by one of the devices driven by that belt. Then it’s a matter of rotating each device, feeling for any play and listening for any sound. The rough or noisy component is suspect #1.
If the tensioner and belt are original, it’s a good idea to replace them anyway.
Until one of you guys sends him an email, he’ll never know the answer.
As of today, May 31st 2024 the noise remains… so. My wife is primarily concerned with getting rid of the noise. I’ve finally got a little money to put into really getting to the bottom of the cause of the noise, … really trying to avoid a car payment.
The noise maybe coming from the timing chain cover area, VVT stuff (sorry mind went blank)… My 2009 has been making noise only on cold start up for 100K miles now, one day I will get around to repairing it… lol
Suggest to clarify the conditions you hear the noise? Do you only hear it while the engine is cranking? Cranking means that ‘rrr rrr rrr’ sound you hear from the electric starter motor when you first start the engine, but before the engine actually starts and begins to run on its own. Or do you hear the noise even after the engine starts? If so, do you hear it when idling in the driveway? If so, how long do you hear it while idling in the warm-up phase before it stops?