Distortion or Clipping does NOT damage the speaker.
There are 2 causes of speaker damage: Mechanical and Electrical.
Mechanical: Physical damage to the speaker due to poor speaker design (like improper crossover points, improper enclosures, etc.).
Electrical: Speaker damage is caused by too much VOLUME which causes too much POWER to the speakers. If the power is not enough to damage the speaker, no amount of clipping or distortion will damage the speaker.
Just don't play anything too loud.
What you say is perfectly correct, but it is not the whole story. In the context of the original question we are really talking about tweeter burn out and therefore power to the tweeter.
Now there is a huge difference between digital and analog clipping. Analog clipping like tape saturation or the amp running out of gas, starts gently and ends up with amps being limited in total power output by the limit of the amp to increase the power.
Digital clipping has sudden onset and rises very quickly. The other issue is that in a badly recorded file the distortion is already in the program and will not be limited by amplifier power.
So lets take two files of the same musical selection. If one file is clean and the other has 10% distortion which is conservative. With digital distortion it can and does go much higher than that.
Now pretty much all the increased power demand will go to the tweeter. So distortion products are up by 10 db if the first file was clean. This means you will see tweeter failure at half the perceived loudness playing the distorted file compared to the clean one. This is because the tweeter is only a small part of the total sound output. The tweeter is the most fragile but will be taking the brunt of the increase in power from the distortion.
This form of distortion is actually much more pernicious than amp clipping which is what we are usually talking about.
So those dealers are essentially correct you have to be careful what you pull down from the Net.