"... in most cases, the people using pirated software would not pay for it if it would be not be possible to use it without buying."
In economic theory, the demand curve is (basically) constructed from the willingness-to-pay (WTP) of people. It is assumed that people with a WTP > Price (P) buy the good, and those with a WTP < P don't.
Your statement would be true if we could determine that only people in the WTP < P group use illegal copies. They would not affect a supplier's sales when they use illegal copies.
But there's no causal relation between one's WTP and one's ability to use an illegal copy. Thus, people in the WTP > P group are just as likely to do so as people in the WTP < P group.
Well, this is not quite right, for we may ask: Who's more likely to be motivated to make or search for illegal copies? Who's gaining the most profit by investing time and effort to search file sharing networks? Those with a rather high WTP or those with a rather low WTP?
If this sounds uncommon to you, ask yourself: What software is more likely to be found in any file sharing network: The popular ones -- that is: those with many people having a high WTP -- or the unpopular ones? According to my observations, the popular ones are more likely to be found. The same holds true for other products such as movies and music.
Whether these losses are compensated by advertising effects, is a matter of the circumstances: To make this reasonable, a supplier needs to sell (more) copies. People with illegal copies may advertise a product, but they are also likely to distribute illegal copies: What good is advertising if it does not increase sales?
Even if we assume that -- in the beginning -- only people in the WTP < P group distribute illegal copies and advertise them, it should be clear that it will also spread to people in the WTP > P group. The longer illegal copies are around, the more likely it will affect the sales of the suppliers.
Overall: Yes, it's possible that illegal copies do no harm to suppliers and may even help them to sell more legal copies. But these are likely to be exceptions to the rule. In general, using illegal copies decreases the wealth of suppliers.
In the long run, it also decrease the wealth of users who want or need software, for it decreases the incentive of suppliers to improve their products or invest money to make new ones.