People tend to confuse the uncertainty principle and the observer effect. It’s true that you can’t make a measurement without disturbing the measured system – but that’s not the reason why you can’t simultaneously measure the exact position and exact momentum of a quantum particle. You can’t measure them because a quantum particle doesn’t have an exactly defined position and an exactly defined momentum at the same time, and that’s what (that version of) the uncertainty principle says.
As for why scientists are convinced that a quantum particle doesn’t have a precisely defined position and momentum, it’s because this is what quantum mechanics tells us, and quantum mechanics is a theory that makes many predictions which are confirmed by experiment. Essentially, quantum mechanics claims that a quark (for example) isn’t really a classical particle, but instead is something more like a wave packet.
Think of a wave extending forever in both directions. It has a precisely defined wavelength, and thus a precisely defined momentum. (In quantum mechanics, momentum is just Planck’s constant divided by the wavelength). However, it doesn’t have a defined position (since it extends forever in both directions.)
You can add several waves with different wavelengths together in such a way that their sum becomes more and more bunched up with each wave you add. (See the link above, and this one). Thus, its position is more and more precisely defined. However, its wavelength gets less defined, since you keep adding waves with different wavelengths.
If you could add infinitely many waves together, this “wave packet” could be made so tightly “bunched up” that it would look like a single spike. (Such a spike is called a Dirac delta function.) Now the position is precisely defined, but the wavelength is completely undefined (since the spike is the sum of infinitely many waves, each with a different wavelength).
So you can either have an exactly defined wavelength/momentum, but a completely undefined position (the case of a single wave with fixed wavelength), or you can have an exactly defined position but a completely undefined wavelength/momentum (the Dirac delta function), or you can have something in between, where there is some uncertainty in both position and momentum (the wave packet). But you can’t have an exactly defined position and momentum at the same time.
Quantum mechanics (which, as I said, is well-confirmed by experiment) tells us that quantum particles behave like wave packets, and thus have this property of uncertainty. (In fact, even macroscopic objects have it, but the effect isn’t noticable because Planck’s constant is so small.) And it is uncertainty of this sort, not any claim that “the observer affects the observed” which is the essence of the uncertainty principle.