Does removing anti-skating really improve sound?

You can prove this to yourself with any good test record like the Hi-Fi News Analogue Test LP. Play the Bias Setting band and lift your anti skate weight. The left channel will start buzzing madly. Add a lot more weight and the right channel will start buzzing madly. Get it right and both channels play the test tones beautifully. This is the only right way to set anti skate and it is at best an approximation as the skating force changes with groove speed and degree of modulation. Any setup that sounds better without anti skating was not set up correctly to begin with.

I'm glad someone else has mentioned Hi-Fi Test LP finally. 
Because people normally posted something like "adjust by ear or using CD or blank record" 

Hi-Fi Test LP is a must have, not only for anti-skating, but for eoverall setup, there is a free protractor inside. 

Why don't readers here just remove then restore a/s to hear if you like it or not

Stringreen, the most important thing I have learned in this passionate hobby over the last 55 years is, Never Trust Your Ears. The problem with ears is that they are connected to a brain. Brains can accommodate to stimuli in amazing ways. Example. You walk into a room with a really bad odor. In about 5 minutes you notice it is not near as bad but then another person walks in wrinkles their nose and says "boy it stinks in here!" 
Always follow the science when you have it. Cartridges in pivoted tonearms track much better and have much lower distortion with anti  skating set correctly. Setting up a turntable correctly is not easy. Very few dealers have someone who knows what they are doing. I hate to say this because as a company McIntosh's customer service has no equal but I just had to re adjust one of their MT5 turntables. Not only was it out of alignment but the resonant frequency was too high. I had to add 2 grams of lead to get it down to 10 Hz. It was supposed to have been set up at the factory. 
Every vinyl jockey should have the tools and learn how to do this to get the best out of their turntable. These are, A protractor ( I like the DB Systems best), a pocket mirror, a bright light, a little engineers square, little screwdrivers, cartridge weights, a level bubble and a good test record like the Hi-Fi News Analogue Test LP.

Oh, I forgot to add. You do not need a Fozgometer or an oscilloscope to do this right with modern cartridges. A good eye and the right tools can do this perfectly. Back in the day cartridges were all over the place. Cantilevers set in crooked, styli not perpendicular to the cartridge body not to mention internal inaccuracies. But today I have not seen a single
Ortofon, Clear Audio, Lyra, Koetsu or My Sonic Lab cartridge that wasn't dead on. Don't forget you have to use cartridges that match the effective mass of your tone arm. You can always add weight to a stiff cartridge but it is much harder to subtract weight from your tone arm when you use a highly compliant cartridge. 

mijostyn

... Never Trust Your Ears. The problem with ears is that they are connected to a brain ...

We'll have to disagree on this. I consider the brain an advantage, not a disadvantage. The ears and brain work in tandem.

Always follow the science when you have it ...

And for that, you need a brain.