You'll notice the tone control working with either a 0.022uF or a 0.047uF cap, or anything in between. It's more a case of how much treble they roll off when turned pretty much all the way down. 0.047uF is the 'standard' selection for single coils as they are normally very bright sounding so the extra capacitance takes off more treble. So you'd notice the effect more than with the 'standard' humbucker 0.022uF selection. not less.
I don't know if you've got 500k or 250k tone pots. and you may have a linear taper tone pots. Audio taper gives a quicker response to turning down the tone control than a linear taper. As the tone pot is used as a variable resistor, (not as a potential divider like the volume pot) a 500k pot will take more turning down to get the same tone roll-off than 250k pot e.g. with a linear pot, a 500k pot on 5 is the same as a 250k pot on 10, so you should be able to see how all the tone action bunches up to be within the very bottom end of the pot's movement with a 500k linear pot.
So these days I prefer to use 250k audio taper pots for tone, for both single coils and humbuckers. You'll lose a very small amount of top end using a 250k tone pot on a humbucker, but it is barely noticeable to me (YMMV). I also normally fit 0.015uF caps, as I never want the full mud sound that a bigger cap value can give, which helps give a more controllable and useable tone roll off over 90% of the pot travel. You'll always get a lot more of the tone roll off action right at the bottom end of the pot's travel, but without a custom taper, there's not a lot you can do about that.
There may of course be a wiring fault or a bad ground connection. But if the tone control only really has any effect down at 3 or less and then the sound is then really dull, then it's more likely to be down to the pot value and the taper type.
I revisited the wiring briefly last night and it all looks OK, so I suspect you've hit the nail on the head.
The guitar has a fair bit of work left before I'm happy with it, but I'm not going to do anything with this until I get around to doing the pup swap. When I do that, I'll bring over the LP's stock pots.
Progress shots:
The hole refill is better but not perfect. I need to fix the action - I noticed some funkiness around the 15th fret on the 2nd string. If I bend up, it's dead. I can fret the 3rd on the same fret and it's fine. Obviously, I also need to trim the strings :P