99% of the time I just have the tone pots maxed out all the time too but now and again I like to roll the tone down a bit when soloing on the neck pickup. I tried it after reading that Slash did this quite a lot and it can be quite effective, though of course it's easier for him as he has a tone pot for each pickup and I don't.
Pretty much this.
I am however using my tone pots more and more. Part of this is upgrading pots and caps (I now use orange drops), but the main part for me was switching to 50s wiring.
After that I really started to discover the tone pot as it worked in way I seem to enjoy more (same with the volume pot, it is not just a on off switch anymore).
Granted I so far only use it for melody work and not for chord riffing (this may follow sometime though) and I find it very cool on the neck to get that extra smoothness which can be awesome for shredding and slow work alike, for the bridge to tame it a bit when soloing if I want that flavour without going too far off from the neck in terms of thickness and brighness. For the middle in a HSH it is a lifesaver sometimes to get it a bit back and when you go to a crunchsound and roll the tone on the middle off a tad you can get the most singing melody tone I have yet gotten from a PU I must say. Incredible really when you hit it (and you can learn how to).
Plus for clean tones it can do so much too, not just if you wanne add that tad of a jazz flavour in there.
Now of course it does not have to work for everyone, but I can recommend to at least try different caps (values and types, experimentation here can be very inexpensive for starters and if you like the results you can always try more expensive PIO caps and such) and 50s wiring.
If you explored it and tried it in an honest way, sure bypass it and all, but maybe give it a proper chance first, you might discover a whole new layer of tone.