But we don't know the actual amount of people which were infected in Italy, because some of these (a sizable percentage) only reported fevers, and perhaps didn't even think about having SARS-CoV-2 in their bodies.

An acquaintance's partner suspects having got it, because after two days (two days, not weeks) of very mild fever (~37.5C) he was hit by anosmia. An ex-coworker also mentioned "a horribly strong fever" with respiratory difficulties which lasted just a few days. Yes, anecdata, but shows that you can easily miss a large part of the infected cases if you only test those hospitalized.