If that were even nearly true then.... I can't think of anything to say there.^No, it isn't. You will achieve a MUCH bigger variation in tone by changing your amp than you will by changing your guitar. A Strat on a Fender Twin is going to sound more like a Les Paul on the Twin than another Strat on a Peavey 5150.
And actually, it's usually not the pickups that make a guitar sound bad...it's the pots. And even then, there's not a huge difference. Most people could not tell the difference between TS's guitar and a real Strat. The correlation between price and tone is very, very poor. What you pay for with a more expensive instrument is generally 1-build quality, 2-place of manufacture, and 3-the sticker on the headstock.
Now that's not to say there aren't some models with crappy pups...microphonic pickups, or those made with really inferior parts, can sound pretty bad. SC/Squier is usually pretty good about it though.
Rest assured that you can put whatever eq you want. Use transistors or valves, add as much distortion as you want, you aren't easily going to make a fender or fender copy with single coil pickups sound like a Gibson or Gibson copy with dual coil hum bucking pickups.*
I suppose in the same way that a person can be auto tuned to the correct pitch and made to sound like a robot you can do anything you want with a soundwave pretty much. But it seems like a lot of trouble to go to. (you're not going to get that great of a sound difference just by changing amps. If yu want a fender strat to sound like a Gibson les Paul for example you'll need to do some pretty major signal processing, not just change amps).
You seem to be under the impression that all pickups are somehow equal, or only divided into good or bad,*
You're forgetting that the thickness of the coil wire, the amount of turns, the distance of the coils to the poll pieces, the material of the poll pieces, the uniformity of the winds (or some believe lack there of). The overall construction (I.e how it's set and what it's set in). Whether it's a single coil design. Or whether there are two coils of opposing polarity spaced closely to each other conneceted together such that and hum picked up in the first coil of the pick up will also be picked up in the second pickup (at rougly the same amplitude frequency and phase). Which wll mean the two noise signals are combined and the noise (hum) induced in the pickups will be reduced (the hum will be bucked) -which is why they are called hum bucking pickups, or humbuckers for short. Also the fact that they have twice the amount o winds as a normal single coil pickup usually means that they are 'hotter' and produce a louder fatter sound.*
I agree the pots used in the guitar can have an effect on the amount of noise a guitar produces (I mean hum and inteference). But I'd hardly say that it has a greater effect on the sound than the pickups. In fact I'd go as far to say that the pickups and amp are 99.9% of the sound. Whilst the pots. Grounding and cables form less than 1% together. And any noise generated from there parts really just means that you need tm fix the guitar. (switch cleaner or replace parts) these tend not to be the parts that even budget manufacturers save on though. Even a quality pot is really low price. So if it comes from the start with one that crackles they aren't going to sell the guitar are they?
Either way. The setup problems that the op is getting with the setup of their guitar really does just mean that the guitar needs to be set up better.
As I said before. Even if you buy a brand new (insert your favourite brand here) guitar it'll still need to have all that work done on it just the same as if you bought a (insert your favourite vintage guitar here) or a (insert your least favourite budget guitar here)*
Just if you bought vintage or crazy expensive new you'd hope that the shop would set it up for you.