Just an FYI, the belt you're referring too is the serpentine belt, I have never seen a sticker for the serpentine belt being replaced nor a sticker coming with a new serpentine belt to be place in the engine compartment, now on the other hand the "Timing belt" should always come with a sticker showing when it was replaced.
A serpentine belt is not as critical as the timing belt, if a serpentine belt breaks it will not usually damage much else for the most part unless it runs the water pump in which case you should shut down the engine until it's replaced, but if the timing belt breaks on many engines it will destroy the engine which is the case on the 4runner V8.
That being said if you don't see a recent sticker for the timing belt (should be replaced every 90,000 miles) or have a record of it being replaced I would be very concerned and get it replaced.
I looked at the video again and was able to hear the audio much better, I still couldn't hear any ticking/tapping noises, but I could hear some whining noises, which could be any of the moving parts, idlers pulleys/tensioner/alternator/power steering pump, etc. being turned by the serpentine belt making those noises, inspect and replace any or all if needed.
I also highly recommend as mentioned to use a mechanic's stethoscope to pin point noises in the engine compartment, you should easily be able to locate the ticking/tapping and whining noises with one, they are cheap usually around $10 or less at any auto part store, everyone should have one in their tool box!