Just because I have only been a member on this forum for one year doesn't mean that I have only supported Juventus for the same amount of time.
Let's get a few things clear. Performing well in Europe doesn't necessarily mean Serie A will get stronger. Other factors like stadiums, youth development, not to mention how well individual clubs are being run economically are more important. However, this doesn't mean that the performances of Italian teams in Europe don't matter. First of all, Europe is a great showcase for Italian football. Fans who otherwise don't watch Italian football might be interested by seeing Juventus destroy Chelsea or by watching Lazio give some good performances in Europa League. This could potentially lead to more popularity for Serie A and more viewers as well as more sold merchandise. People who otherwise thought Serie A was a boring defensive league might chance their mind. While I don't care about these individuals personally, more viewers benefit the league financially. Seeing that we are able to defeat a team like Chelsea will also attract players. It proves that our league maybe isn't that bad after all and that we really are a competitive and interesting team.
Good performances in Europe are also important when it comes to the UEFA rankings. 4 Champions League spots are crucial. Fiorentina would have a much easier job keeping their biggest players (Jojo) and attracting new ones if the 4th place actually gave a Champions League spot. Good players want to play Champions League football - it's as simple as that. A Champions League spot is of course also crucial financially.
Let me put it this way: Do you think it doesn't matter whether or not Italian teams perform in Europe?
What I simply meant by my statement is that if you know Italian teams are playing in Europe and don't care about their performances, it doesn't make you a better fan. It means that you either don't think their performances in Europe matter or that you simply don't care about Italian football in general. If you think the latter and still don't support Italian teams in Europe, I simply don't get it. Of course there is also a third reason; hatred. I would never support a team like Inter because of the history, but I don't see why you wouldn't support Lazio in Europa League for example (I'm not going into the stupid nazi debate btw...).
Good Europa League or Champions League results probably won't save Italy, but it could at least help stopping the huge drop in quality just a little bit and give some capital injections to the clubs.
I wrote that post 1 AM as the last thing before I went to bed, and English isn't my mother tongue. Furthermore, I actually used the right word further down in my post. Do you really think it was necessary to correct me?
I apologize about potential mistakes in this post in advance.