As the buzz comes from taking your hands off the strings, that shows that the strings are grounded properly. Your body is being given a good connection to ground, and either a) any electrical noise your body is picking up/giving off is taken to ground so it doesn't affect the pickups or b) your body now acts as an effective soak for a lot of the EMI floating around (or a mixture of both).
Shielding the pickup cavities isn't standard for a Les Paul (though it's a good idea for those fitted with P90s). Gibson keep changing their control cavity arrangements, but on my very new 2018 LP Classic Goldtop (with P90s), they've got the pots connected to a metal shield plate (like they have done in the past), which helps. they've also added a small tag board, which minimises the soldering they have to do to the back of the pots.
With covered pickups with shielded cables, there's not a lot of point in adding more shielding to the pickup cavities. Even with uncovered humbuckers, you normally don't get enough noise problems to make shielding the cavities necessary. Also on most guitars with non scratchplate mounted pickups, it's near impossible to shield the cable passageways to them, so you'll never manage complete shielding coverage. But every bit of extra shielding will bring the noise levels down slightly, so if you are getting noise issues, then it's certainly worth adding as much shielding as you can.
The noise has to come from somewhere though, so unless you are near to a powerful radio mast, it's worth checking around your house for any sources of electrical noise which can be turned off, especially when recording, e.g. dimmer switches and fluorescent lights.