Actually, the Catholic Church teaches that no other religion but the Catholic Church
can grant salvation. There is no salvation outside of the Catholic Church, so I think that you are wrong to say that other belief systems are “unlikely” to grant salvation, they can’t ipso facto by their very nature. Salvation comes to people outside the Church through their spiritual baptism into the Church through following the Will of God as known to their conscience.
On the other hand, the Catholic Church believes and teaches that the world religions have arose from man’s openess to God and have at their origins the inspiration of the Holy Spirit. The Church believes that there is divinely inspired truth, seeds of the Word, in all world religions. These truths, the Church teaches, are the presence of Christ in these religions and by embracing these truths, these non-Christians are actually embracing Jesus - who is the Truth incarnate - without knowing it. These teachings, which in many cases came from the religion’s founder’s genuine experience of the Holy Spirit, guide the followers of these religions to more faithfully adhere to the dictates of their consciences and therefore to attain to salvation.
In this respect, we actually do not believe that other religions “are man-made”. That is erroneous.
And as to how many non-Christians attain to salvation we cannot possibly know just as we cannot know how many Christians attain to salvation. Only God judges hearts and minds and knows this, so please never assume with your own knowledge to judge other peoples’ salvation. However, the Fathers of the Church seem to suggest that MANY non-Christians are spiritually members of the Church:
“…How many sheep there are without, how many wolves within!..When we speak of within and without in relation to the Church, it is the position of the heart that we must consider, not that of the body… All who are within in heart are saved in the unity of the ark…”
- Saint Augustine, Church Father (354–430 AD), Baptism 5:28:39
“…He was ours [a Christian] even before he was of our fold. His way of living made him such. For just as many of ours are not with us, whose life makes them other from our body [the Church], so many of those outside [the Church] belong to us, who by their way of life anticipate the faith and need only the name, having the reality…”
- St. Gregory of Nazianzus, 18.5 (c. 374 AD)
That being true then one should expect many non-Christians to attain to salvation.
After all God desires all men to be saved.
St Thomas Aquinas said in the 13th century:
“…It is the characteristic of Divine Providence to provide every man with what is necessary for salvation…In the case of a man who seeks good and shuns evil, by the leading of natural reason, God would reveal to him through internal inspiration what had to be believed…”
and Vatican II in the mid-20th century:
“…Since Christ died for everyone and since the ultimate calling of each of us comes from God and is therefore a universal one, we are obliged to hold that the Holy Spirit offers everyone the possibility of sharing in this Paschal Mystery in a manner known to God…”
as well as Blessed Pope John Paul II saying in the late 20th century:
“…The universality of salvation means that it is granted not only to those who explicitly believe in Christ and have entered the Church. Since salvation is offered to all, it must be made concretely available to all…For such people salvation in Christ is accessible by virtue of a grace…This grace comes from Christ…”
And thus Jeff Mirus writes on “Catholic Culture”:
“…People can be saved without explicitly embracing Jesus Christ and His Church. This is not merely my assertion, of course, but the clear teaching of the Magisterium. The decrees of the earlier councils and popes admit the possibility [of salvation outside the Visible Church] by referring to being “added” or “joined” to the Church, rather than insisting upon explicit formal membership, as a condition of salvation (e.g., Council of Florence in 1442, Cantata Domino, Denzinger #714); and this traditional teaching has been developed and carefully explained by, among others, Pope Pius XII in Mystici Corporis Christi (103), by the Second Vatican Council in Lumen Gentium (16), and by Pope John Paul II in Redemptoris Missio (10)…”
That is Catholic doctrine
Rather than speculating about the salvation of non-Christians and indeed trying to claim like your good self how unlikely it is for them to attain to salvation, Catholics should be working on their own salvation.
Because, sure as God, you can bet that “to those who are given more, more will be demanded”. If a Catholic does not lead a just life, having all these graces and the fullness of truth, then he will fair far, far worse than a non-Christian who might be excused on grounds of lack of knowledge.