Old Timer 'Twang' Telecaster pickups
The biggest reason your cheap Tele sounds poor is due to the use of ceramic magnets and steel pole pieces in the pickups. Even in middle of the road Teles you can find such pickups sometimes. These pickups are not made to sound good. They're made to make noise and save money. No Tele pickup was made this way in the early days when they were first designed, they were designed to sound good, with alnico rod magnets.
Here's a set of Tele pickups with alnico magnets. These sit nicely between the ceramic rubbish that you can find stock in many Teles and boutique offerings (such as our exclusive Alegree pickups). These use high end components, but are machine wound to a rough spec. You'll get yourself 70% of the way to a boutique offering for a fraction of the price.
The specs (neck/bridge):
Resistance: 8.5k/7k (different gauges of wire are used - this doesn't mean the neck is more powerful)
Magnets: Alnico 5 rods
Copper plated steel baseplate (a ferrous baseplate is a VERY important component of Tele bridge pickup tone that many pickups in this price range lack)
Chrome plated brass neck cover (vintage material)
Cloth covered output wires
Supplied with mounting screws & springs.