Why would God's covenant people think that they should baptize their children as a "sign of the covenant God has made with them." (Note that I'm not sure if the antecedent of the last word "them" refers to the children or to the parents.)
Of course, it is entirely logical for adult believers to be baptized since, assuming Calvinistic soteriology, their faith is evidence of their election and inclusion into the covenant. But what makes them think that their children should be so baptized. No one thinks that just because the parents are elect the children must therefore also be elect.
I am currently a bit of an oddball on this issue because my understanding of baptism is covenantal but I lean toward credobaptism. But I'll give you my understanding of it, and yes it is a very complicated issue, and before you can accept the Reformed position on paedobaptism you must first have a Calvinistic soteriology and also a covenantal understanding of how God relates to his children.
So bear with me as we go back to Romans 9 and look at how God relates to his children from a Reformed perspective. I understand, as an Arminian, you will disagree with what I think Romans 9 says, but we cannot discuss Reformed paedobaptism unless we're first clear on how the Reformed understand Romans 9 and how it relates to God's covenant. Starting at the end of chapter 8:
38 For I am sure that neither death nor life, nor angels nor rulers, nor things present nor things to come, nor powers, 39 nor height nor depth, nor anything else in all creation, will be able to separate us from the love of God in Christ Jesus our Lord.
1 I am speaking the truth in Christ—I am not lying; my conscience bears me witness in the Holy Spirit— 2 that I have great sorrow and unceasing anguish in my heart. 3 For I could wish that I myself were accursed and cut off from Christ for the sake of my brothers, my kinsmen according to the flesh. 4 They are Israelites, and to them belong the adoption, the glory, the covenants, the giving of the law, the worship, and the promises. 5 To them belong the patriarchs, and from their race, according to the flesh, is the Christ who is God over all, blessed forever. Amen.
6 But it is not as though the word of God has failed. For not all who are descended from Israel belong to Israel, 7 and not all are children of Abraham because they are his offspring, but "Through Isaac shall your offspring be named." 8 This means that it is not the children of the flesh who are the children of God, but the children of the promise are counted as offspring. 9 For this is what the promise said: "About this time next year I will return, and Sarah shall have a son."
As you well know, Paul has just finished saying at the end of chapter 8 that nothing can separate God's people from him. Then he is confronted with a gigantic problem. If nothing can separate God's people from him, why are so many Jews rejecting the Messiah? God made a promise to the Israelites to send the Messiah to redeem them. God sends the Messiah, Israel rejects the Messiah, and Paul is faced with a huge problem: Israel, rejecting the Messiah, is now accursed and cut off from God. But in chapter 8, Paul has just finished saying
nothing can separate God's people from him.
This is massive. You can see in verse 6 Paul is struggling with the problem that it looks like God's word has failed. But Paul's solution to the problem is that God's word has not failed because he didn't make the promise to physical Israel but instead to spiritual Israel. "But it is not as though the word of God has failed. For not all who are descended from Israel belong to Israel. . . . it is not the children of the flesh who are the children of God, but the children of the promise are counted as offspring." God's promise does not apply to the physical offspring of Abraham but the spiritual offspring of Abraham.
This is how God works all throughout the Bible. He makes covenants with his people, his people break their end of the covenant, and God graciously upholds his end of the covenant by performing their end in his Son. "If we are faithless, he remains faithful" (2 Timothy 2:13).
So the covenants of God are massively important to us because we believe that God's covenant with us is the only assurance we have. His covenants are conditional, but he himself has performed all of the conditions, so we can be certain that nothing can separate us from the love of God that is in Christ Jesus our Lord.
Our understanding of baptism is that it is a sign of God's covenant with us. We do not see baptism as an outward expression of an inward faith, but instead as the exchange of covenant vows: We are baptized in water (our end of the covenant vow) and God baptizes us in the Holy Spirit (his end of the covenant vow). And the covenant we are a part of is the covenant God made with Abraham that through Jesus -- a physical descendant of Abraham -- all who are spiritual descendants of Abraham would be counted righteous.
So since we understand baptism as a sign of the covenant, and we also know that circumcision was a sign of the covenant God made with Abraham, you may be able to see why it's common for the Reformed to baptize their infants as a sign of the covenant, just as Jews circumcised their male children as a sign of the covenant.