Tensioners can cause problems at certain speeds / engine RPMs - just depends on what is wrong with them. Though the tensioners seemed to be more of an issue with the 9th gen than the 8th gen Corollas.
Disconnecting the VVT-i solenoid results in no pinging = interesting.
That means that either there is a heavy build up of carbon inside the combustion chambers (taper squish design is more prone to this, causes the cylinder to rock slightly within the bore), the VVT-i solenoid is smoked, the cam sprocket is damaged, or the ECM is getting incorrect sensor readings from any number of sensors (bad sensors or bad ECM).
Have you tried filling with premium gas and see if the noise goise away or not? If it does then I would lean toward a carbon buildup issue. Otherwise, disconnecting the solenoid would show that the car doesn't run well with a specific valve timing/overlap - that could point to many potential issues.
First, lets try and verify that carbon is not the major cause. This can done with regular or premium gas in the tank, though running with premium - you can knock off two items to check on your list at once. The 1ZZ-FE engine unique piston design helps with combustion efficiency, but also makes it much easier to foul up with carbon deposits at lower engine speeds. One quick way to burn these off is to run the engine at WOT at highway speeds.
Easiest method (highway) is to run the car at a steady 45-50MPH with O/D switched off or even in 2nd gear (4-speed automatic) of 3rd gear pulls with a standard 5-speed manual. Then punch the gas and accelerate to 60-65MPH+ (no need to go crazy, no need to risk a speeding ticket or reckless driving here), then let it coast down to 45-50MPH via engine braking. Repeat several times. This will help blow loose debris right off the tops of the pistons, and will pull enough vacuum to help clean the rings as well. Should be part of every engine break-in procedure - both of my current cars have seen this, all my cars have had this over the years and all performed well.
For the solenoid and VVT-i sprocket assembly - will need more equipment and time to address those. My suggestion is to take it to a well respected dealership and let them run the test on it for you - the amount of time and labor going into it will be worth the $85 diagnostic fee, IMO. They will be able to run test with a Toyota scanner - enabling and disabling the solenoid and sensor readings electronically and seeing how the car reacts.
As for additives, I think at this point, the engine actually sounds pretty clean - could be a case of not driving the car hard enough. Sounds kind of strange - but I've seen weirder things happen. Used to work on older GM Northstar engines, those showed similar behavior, just took them out for a couple of WOT apps - 8 out of 10 times that fixed the issue. This particular engine is also very sensitive to electrical noise - if any additional electical load was recently added, it could cause weird issues in the sensors and ECM. Poor engine/chassis grounding could also be a possibility. Also, this engine has a 10:1 compression with a very tight piston to head tolerance (taper squish) - 1ZZ-FE engine is known to be very noisy and have certain "features" like a weird stumble at 3500-4000 RPM in some cars, tapping and rattling at startup and idle, and lots of piston slap in cases of heavy carbon buildup. Lots of times, motor oil choice can minimize or aggravate this behavior. Good example is Royal Purple in the 1ZZ-FE/2ZZ-GE engine family - many have found that the engine sounds like someone poured a bunch of nuts and bolts into the engine. Can be scary sounding - but the engine oil still did its job and UOAs looked clean. I've noticed that as well - with Mobil 1 compared to Pennsoil Platinum - the Pennsoil is known to be a little thicker, engine runs well on both, but the PP quiets things down considerably (though fuel economy suffers very slightly).
What's the mileage and type/weight of oil you run on this engine?