This is going to delve into a number of specific points of Calvinist theology, so ... if you're reading based on wanting to know (this is "Ask a Calvinist") be prepared for a lot of subtle detail.
The HUGE problem with this belief is that it makes salvation conditioned on something other than the finished work of Christ alone.
Excuse me? This isn't the case. The covenant they are a part of is based entirely on the finished work of Christ, alone.
If the children of believers are automatically "holy" based simply on the fact that their parents are believers, then some people's salvation is based on their lineage.
Well, your reasoning has some problems to overcome. One problem with this conclusion is that holy people are not necessarily saved. Witness ... Jonas, the prophet to Nineveh. He was holy all right: so holy -- so "set apart" -- that he wanted to set himself as far apart from Nineveh as he could. Holiness is not such a direct indicator of salvation. With "Don't come near me! For I'm too holy for you," Isaiah describes a people who challenge the very holiness of God.
Another problem with this reasoning:
people are set apart and holy in the New Covenant because of their familial relationships. That's simply an explicit, stated fact of the New Testament. Paul pointed out, "For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. Otherwise your children would be unclean, but as it is, they are holy." 1 Cor 7:14 The children of believers are indeed holy because of their believing parent or parents. God said it. That settles it. It'd be great if everyone believed it.
Now, the Synod of Dordt does project this holiness onto the household covenant that God made with Abraham as well as with David and with us, pointing out that there's no warrant to doubt the ultimate election of children in our household who haven't reached a point where their actions would show the direction of their lives. That's true. We don't think small children are all condemned to Hell in their unawareness until they embrace Christ. We certainly don't believe such a thing. Do you? And if not, on what basis would you make that claim?
This is exactly what the Pharisees, etc, believed.
They also believed God was good. Should we stop believing that, because the Pharisees believed it?
Certainly the Pharisees were wrong in believing that lineage directly concluded their salvation (and I doubt seriously the Pharisees actually believed such a thing, btw). But they were not wrong in believing that the children of believers are holy. Paul again says, referring specifically to unbelieving Jews, "As regards the gospel, they are enemies of God for your sake. But as regards election, they are beloved for the sake of their forefathers. For the gifts and the calling of God are irrevocable." Rom 11:28-29
Pharisees would not be wrong that the children of believers are specifically in view as regarding the covenant's promise, either -- that's what Peter said in Acts 2:39: "For the promise is for you and for your children and for all who are far off, everyone whom the Lord our God calls to himself."
They believed that because they were the children of Abraham they were saved; however, the apostle Paul in Galatians 3 clearly explains who the children of Abraham are, and it's not based upon one's lineage.
This is accurate. It's just not pertinent. "What advantage does the Jew have over the Gentile?
Much in every way." Rom 3:1ff "Theirs are the covenants ..." Rom 9:4ff, talking about the Jewish people, whom Paul laments
do not believe in the main. So being parties to the covenant is indeed preferentially based on children in the households of believers. This isn't rejected by the New Testament (and I've quoted the New Testament to make this point). However, the New Testament makes it plain that the children of Abraham are children of the promise -- the promise to Abraham explicitly includes Gentiles way back in Genesis 22. So that expands the meaning of being a child of Abraham. But it doesn't remove Abraham's physical lineage from the bunch (cf. Rom 4 as well, which picks through this point in fine and subtle detail).
Being of the physical lineage of Abraham never was a direct implication of salvation. The Pharisees didn't really believe that, as their view of "tax collectors and sinners" demonstrates, both of whom included children of Abraham.
Furthermore, this belief stands in opposition to election. God said in Romans 9:10-13,
"10 And not only this; but when Rebecca also had conceived by one, even by our father Isaac;
11 (For the children being not yet born, neither having done any good or evil, that the purpose of God according to election might stand, not of works, but of him that calleth)
12 It was said unto her, The elder shall serve the younger.
13 As it is written, Jacob have I loved, but Esau have I hated."
Esau was an unsaved person, who had parents that were believers. God has elected His people before they were yet born, and their parents beliefs have absolutely nothing to do with it.
So Esau was not party to the covenant he so clearly was party to, because Esau was not chosen to salvation? I say, No. Esau was party to the covenant. If Esau were not saved as you say, then he fell from that covenant. If Esau were saved, then the choice God makes in 9:10-13 is not specifically salvation. =Shrug.= Either view is permissible. Paul isn't necessarily describing God's salvific/reprobative choices throughout Romans 9. Paul is describing the characteristics of God's choices in Romans 9. But even if Paul were describing the ultimate reprobation of Esau, no one denies Esau was party to the covenant through Abraham. Denying his covenant with God is one thing. Being in covenant with God is rather prerequisite to denying that covenant -- otherwise, covenantal denial would be logical and reasonable, because it would be the reality of the situation.
