Is this noise present when using a lot of gain and/or pedals? If so, I wouldn't expect miracles- when someone describes a pickup as 'noiseless', it's because it was noiseless when tested under specific conditions, not in ALL conditions.
But the pickups might not be what's allowing the noise to enter- wires that aren't shielded or twisted are a prime reason something may be noisy and this applies to hum AND buzz- it's important to find out where and how the noise is entering. Some hum is caused by ground loops, which are usually caused by some device (like the amp) being on one outlet and something else (like a pedalboard) powered by another outlet or on a different power phase, which means they're each using one of the two Hot cables entering the electrical service panel, but they're not using the same one. The loop is caused by resistance between the devices and it could be on the hot, neutral or ground.
Hum can also be from the magnetic field created by transformers- small devices don't create a strong one, but an amplifier can if the shell of the transformers is loose or a cable is close to a high voltage conductor in the amp. This is one of the reasons people comment on the wire routing in amplifiers.
Buzz is usually coming from after the rectifier and before the filtering and in my case, it comes from my TV. When that isn't on, by Offset Tele is extremely quiet. For the record, it has the original pickups, AFAIK- they weren't changed before I got it and the photos on Reverb look the same. I just don't understand why they didn't ground the neck pickup cover (the Reverb photo shows it the same) and they didn't use a metal plate under the bridge pickup which has always been grounded on all of the bridge pickups I have seen. 62 years later and they still aren't throwing us much of a bone.
When I first brought this guitar home, it was noisier if the TV was off, so I twisted the pickup and jack wires & added a ground wire to the neck pickup cover- I connected the cover to the black wire but it wasn't very effective. Using a separate wire will also make it easy to replace the selector switch, if I want to add other wiring configurations, such as a series or out of phase position.
Since the third wire and twisting helped but weren't totally effective (in my weird mind), I used Copper foil in the cavities, soldered at many places where it overlapped (but not all- that will be done next time I change strings). I also installed a four wire terminal strip, to make changing the pickguard/neck pickup easy and to allow me to use a piece of thin shielded, twisted pair Belden cable to connect to the controls.
Coming from someone who has worked in Audio/Video, home theater and network systems, I hate noise. It's not always easy to remove and that's something people who use systems other than musical instruments, specifically electric guitars, amplifiers and effects, do not tolerate. If the frequency response is changed by removing the noise, it can be adjusted in some way to recover that but if someone is playing and they stop touching the metal parts on the guitar, even at bedroom level, they shouldn't have to put up with the sound of a chainsaw, IMO.
I'll see if the noise is there next time I see Greg play.