Actually the opening question is facetious. I do in fact know why Catholics today think 'No Salvation Outside The Church' is an outdated teaching. It is multifaceted.
First, most parents don't know their faith anymore because they are victims of the 'love and parachutes' era of CCD. (See my post about Female Acolytes and Vocations for an explanation.) As a result, they don't know anything from basic prayers to tenets of the faith. They cannot teach their children their faith, and we are more than 40 years into this insanity.
Second, the ecumenical dialogue has degenerated into Catholics participating in 'Tent Revival Meetings' with ecclesial communities, where priests are featured speakers (sans Roman collars) and it is covered on the front page of the diocesan paper.
Third, to modernism, indifference, and egotism, the Catholic church takes a back seat. How can the Church get through to people who put sleep ahead of Holy Mass? People are so far from the sacraments, they don't think they need confessionals any more. No one sins these days! When you try to enlighten them, you are threatened or attacked (hopefully just verbally and not physically) but it is harder and harder to evangelize when you must teach your Catholic brethren their faith too!
If people knew their faith, really knew their faith, they would know that three Ex Cathedra statements have been made regarding No Salvation Outside The Church.
Pope Innocent III and Lateran Council IV (A.D. 1215): "One indeed is the universal Church of the faithful outside which no one at all is saved..."
Pope Boniface VIII in his Papal Bull Unam Sanctam (A.D. 1302): "We declare, say, define, and pronounce that it is absolutely necessary for the salvation of every human creature to be subject to the Roman Pontiff."
Pope Eugene IV and the Council of Florence (A.D. 1438 - 1445): "[The most Holy Roman Church] firmly believes, professes, and proclaims that those not living within the Catholic Church, not only pagans, but also Jews and heretics and schismatics cannot become participants in eternal life, but will depart `into everlasting fire which was prepared for the devil and his angels' (Matt. 25:41), unless before the end of life the same have been added to the flock; and that the unity of the ecclesiastical body is so strong that only to those remaining in it are the sacraments of the Church of benefit for salvation, and do fastings, almsgiving, and other functions of piety and exercises of Christian service produce eternal reward, and that no one, whatever almsgiving he has practiced, even if he has shed blood for the name of Christ, can be saved, unless he has remained in the bosom and unity of the Catholic Church."
What more remains to be said? Pray for Catholics everywhere. Pray for all people everywhere. Pray for our Pope! Thank you, God, for such a holy man as Benedict XVI. I think he has a lot more to say, the liberals are quaking in their boots.