I've put 4-way tele switches on several guitars without any problems like that. Thoughts...
#1. If you temporarily reconnect the neck pickup case to the terminal you disconnected it from does that change anything? Not that it should... (Obviously you would avoid the series linkage when you were trying this.)
#2. On Fender and Squire teles ground wires are soldered to the backs of the two pots which then connect to the metal plate through the metal pot bushings and the frame of the selector switch is grounded to the metal plate through the 2 screws.
#3. On the original non-commercial GuitarNuts site back in ~1997 there was an article about keeping the signal returns from the pickups isolated from the grounds until they were joined together at a single point. Until I read that I had been treating the signal returns and grounds interchangeably on my custom wiring harnesses figuring that the less wire used the quieter the guitar would be. Wrong.
After reading the article and implementing some of the ideas my guitars usually got quieter. My theory was that the various grounds pick up crap from what I've always called "the ether" and by joining them all together some of that crap gets cancelled out. By connecting the signal returns to the grounds in a single point you might end up with less crap from the "ether" on the output signal from your guitar. (This is not exactly the same as star grounds on amps which have to with current flow on the chassis.)
#4. Many of my Asian guitars come with conductive paint inside the control cavity and quite often the pickup cavities as well which my US guitars usually don't. I'll go through my guitars and do everything on my OCD list to minimize the noise like shielding the cavities and sometimes using shielded cable on Fender-style single coil pickups (one trick is to wrap an insulated wire around the individual leads and connecting that wire to ground. IMO tedious but effective.)
Good luck exorcizing those darned gremlins!
P.S. I usually wire in a phase switch on a push-pull tone pot since I love the OOP series linkage. On guitars with only a master volume control the parallel OOP linkage can be too thin. On a guitar with individual volume controls you can reduce that thinness by turning down one of the controls a little bit. I finally figured out that you can get similar results on a Tele by adding in a resistor or trim pot to ground from the normally grounded pickup lead on the DPDT switch. (The resistor or trim pot is out of the circuit when the pickups are in phase.)
This idea can also be used on guitars with two volume controls but you will need to decide which pickup you want to attenuate and wire that one to the phase switch. By doing this you can instantly switch to a decent sounding OOP linkage with a single "click."
P.P.S. You can get a better ground on the control plate by wrapping a short length of small gauge stranded wire around one of the screws securing the switch to the plate and soldering it to the back of the pot. (With small gauge stranded wire squished between the switch body and the control plate the switch won't be cockeyed as it would be if you used, say, 18GA solid wire.)
P.P.P.S. I bet that there is not full continuity between the backs of the pots and the bushings because normally that extra wire would not be needed (I only add them if I need to ground a terminal on the switch.)