If you just open your borders because of a drop in population, the country might face bigger problems. There might increase in criminal activities since most of these illegal immigrants don't go through security checks. A handful of European countries are going through insecurity because they were liberal. But letting people who need protection shouldn't stop, even if they are poor.
But the problem with the Italian National team is not a lack of players. Italy produces some of the best players in the world. Immigrants could just be an addition and not the main source for national team players.
That is a different topic altogether. Currently 90% of the immigration to Europe consists of military age men, mostly Muslim, either from sub-Saharan Africa or Arab speaking countries. Given their age group, one would expect them to be extremely productive, and major contributors for sports. But that is not always the case. In case of Italy, there was some attempt to bring members of the Italian diaspora back to the country. The diaspora now numbers around 80 million people, which is around 2x the population of Italy.
Actually the Italian population is 59 millions, not 40, there is a big difference. Anyway immigration just to have more available football players is one of the most ridiculous things I've ever heard. You get good football players having a clear project, having infrastructures, training facilities, good coaches, etc etc. Based on what I read here it simply looks like that a country needs more people, then explain why China and India are so bad that they can't even find 11 good players with more than 1 billion people each...