Notice how during your reasoning the term that is discussed shifts from 'very big problem' to 'a problem' to 'contribute to a problem' and back to 'leading to a problem'. First making the statement small enough so that one cannot disagree (contribute to a problem) and from that feeling of agreement the statement is made bigger again (lead to a problem). This is a dangerous rhetorical tactic that you may have used unconsciously."Immigrants who are in the U.S. illegally are a very big problem for the country today." has a similar issue. "A very big problem" is deemed an opinion phrasing, but this is a statement of fact. The "opinion" here, is that the results of immigrants is a problem, not that immigrants contribute to what was deemed a problem. For instance if overpopulation is a problem, and immigrants lead to overpopulation, then immigration leading to a problem is a factual statement. The end result, "the problem" being a problem, is an opinion. This double meaning likely trips people up.
Everybody contributes to most of the problems in society, including overpopulation. But that is different from 'leading to a problem' and completely different from 'being a very big problem'.
Illegal immigrants contribute to the problem of overpopulation. But their contribution to the total population is small, so they do not 'lead to overpopulation'. And because of this small contribution, their presence certainly is not a 'big problem' in the context of overpopulation. So 'Illegal immigrants lead to overpopulation' is not a factual statement.