We can't have a Church that teaches no salvation outside the Church, and now it does not even matter if you are Christian - Muslim, Hindu, Atheist - no problem.
I would also have to add that I see nothing in Catholic teaching of our time that has ever said that it "doesn't matter" if you are Christian or not.
What the Church has consistently said is that God's desire is to save everyone, and that if someone through no fault of their own is ignorant of the Gospel or Christ and His Church, that in ways known only to Him God can reach that person. That is in no way salvation apart from Christ and is totally dependent upon His merits.
To believe otherwise is to accept a form of Calvinism, which would condemn people to a predestination of hell based upon the time and the place in which they were born. Never had a chance to hear the Gospel? Then obviously you're not one of the 'elect' God chose to save. It would also mean that our salvation is not only dependent upon the merits of Christ but upon the works of others who may or may not choose to share the Gospel.
The Church also clearly states that even though we recognize that God isn't dependent upon our efforts to save someone, we have a duty to evangelize, so yes, it does matter a great deal. It's the normative way that people come to know Christ and are saved.
The concept of invincible ignorance is deeply rooted in Scripture with St. Paul's refusal to judge those outside the Church (1 Cor 5:12-13) along with other passages (John 15:22, Acts 17:30, 1 Timothy 1:13-14).
I will agree with you that in practice many Catholics have an "it doesn't matter" attitude, and I believe that is part of the fallout of the way Vatican II was adopted, and that needs to change. It's our choice if we're going to be part of that change. But I have found no evidence that the Church ever changed its teaching on this.
One of the better things I've ever read on this topic:
Does Being Catholic Make a Difference? | EWTN