1) If it took a miracle, then yes, God would provide a miracle. What you're missing is that no matter who gets baptized into the Catholic Church, whether it be you, me, the infant of good Catholic parents who scheduled it two weeks prior, the catechumen about to die or be martyred, the North Korean who desires it, or via a miracle, all are baptized into the Church via the same providence of God. The only reason that God does not succeed in getting others into the Church must be found in the reluctant will of those who do not enter it.
1)Naturally, that is a fact. However, I don't think you would be able to say that necessarily if someone doesn't believe right now it is their fault. It could also mean that their time has not come yet. Grace is a gift from God, after all.
St. John (1:9) tells us that God has enlightened every man who comes into this world: "enlighteneth every man that cometh into this world."
He also said that those who don't believe in Him, sin: John 16:8 "And when he is come, he will convict the world of sin, and of justice, and of judgment. 9
Of sin: because they believed not in me."This must be accepted as absolute truth whether one is a cradle Catholic or an ignorant native. Why? Because God brings every one of us into this world enlightened for the same reason - to know, love and serve Him. It is toward that end He *will* provide everything we need to get to know, love and serve Him regardless of our circuмstances - provided we do our part. Most people choose, of their own free will, to not do their part, hence suffer for it eternally.
2)The first part of your statement is naturally true. The second part of your statement does not follow. You believe in the error that BOB/BOD don't exist, so naturally you assume that if someone did not receive the sacrament naturally it must mean they weren't true sincere. This seems similar to the error of predestination as protestants envision it.
A BOD / BOB is salvation through faith alone. Accordingly, if one may baptize themself via some presumed desire, one may wonder why the recipient does not simply desire to ascend themself body and soul right into heaven.
Again, the only way a BOD works is when God's Providence is completely taken out of the formula, which is to say that with the divine providence, a BOD fails, without it, it works. With the Divine Providence, God fulfills the desire by providing the sacrament, without God's providence, the person's faith alone saves him - this idea is entirely protestant and condemned by the Church.
3)Of course God controls all events in the universe. However, God has a certain "style" when intervening in human matters. Do you see what I am talking about?
Yes, God's "style" goes about largely unnoticed, almost always His Providence is taken as a matter of natural order, or routine, or planning, like the way He feeds even the birds.
"Behold the birds of the air, for they neither sow, nor do they reap, nor gather into barns: and your heavenly Father feedeth them. Are not you of much more value than they?" Mat. 6:26
"If you then being evil, know how to give good gifts to your children: how much more will your Father, who is in heaven, give good things to them that ask him?" Mat. 7:11
Ever consider how many birds there are - many billions for sure, each one fed by God. That's an example of God's style.