Why are natural marriages between non baptized people recognized by the Church and their children as legitimate?
How can they be married without the sacrament?
I've never heard of a natural marriage - what is it? Wiki seems to be calling a mixed marriage "natural".
Remember that the Church recognizes all marriages as valid - at least insofar as the parties were able to marry ie no prior marriages, legal age etc. The primary (not only) reason the Church initially presumes that all marriages are valid is because She is the only defender of the Sacrament.
As such, She, initially at least, presumes validity of all marriages for the sake of the Sacrament, for the good of the family unit, for society in general etc. whether the marriage is sacramental or not.
Matrimony consists of, 1) the contract between a man and woman called simply, The Marriage Contract. This contract is the thing that is binding till the death of one of the spouses and yes, it happens when the man and woman make their vows to God. This contract, sacramental or not, is what binds. Many people want to believe if a marriage is not sacramental then it is not binding - look what that belief has progressed into - homo marriage..........what's next we can only wonder.
And 2) The Sacrament, wherein the priest blesses the union entered into by both parties to sanctify the marriage, with which comes special graces to help each other put up with each other's weaknesses, in sickness an health, better for worse etc. till one of the spouses die.
Least ways, that's how it was pre-V2.