You should probably look for a different chip and build a sine wave generator (if that's what you want) instead of trying to "fake it" with a 555...
But first, it wouldn't hurt to play around with Audacity and connect the piezo to your soundcard so you can diagnose your problems and decide where you want to go with this.
I tried it. Not much change. I was using 1k resistor and 0.1uF cap. The volume decreased, but not much else
Did you make a low-pass filter or a high-pass filter?
Of course, when you filter-out the harmonics you are filtering-out some of the sound so it won't be as loud. Plus, a simple RC filter is far from perfect...
Those values are probably in the ballpark, but of course it depends on the frequency. Did you calculate the frequency of the 555 oscillator? (I didn't.)
Did you calculate the duty cycle? (I didn't.) A square wave has a 50% duty cycle and will sound the "least-harsh" of any rectangle wave. It's been several years since I used a 555 but from what I recall, it's not easy to get a square wave. And if you adjust the pot, you're probably adjusting the duty cycle so you no longer have a square wave. A type-T flip-flop will convert a rectangle wave to a square wave at half the frequency, but if you want a "pure tone" you want a sine wave.
About a million years ago the company I was working for was designing a new product and they wanted a "beep" for feedback when you pushed the buttons on a membrane keyboard. At first they were trying to get a pleasant sound, and the joke was that they were trying to get the least-annoying sound.