So, then, does the doctrine of irresistible Grace and the "Elect" remove a person's free will?
No, but a full account of this gets complex.
No one's will is completely free. There are external constraints. I am not free to fly to the moon by flapping my arms. I am not free to murder someone, at least not without unacceptable consequences.
Then there are internal constraints. What I can do depends upon my abilities, and even my character. Certain things would never occur to me as possibilities, others might, but still they would not be realistic possibilities.
Christians have generally said that a person can't know God and live a godly life without God first reaching out to them and changing them, or perhaps starting a change in them.
Yet we are free in the sense that for most actions, including having faith, no one is forcing us. That's not how God works, except possibility in unusual situations such as Paul's conversion, and even there he could have responded to the voice in varying ways. So our wills are predisposed and enabled by a variety of factors, internal and external, but we still choose, and most of the time that choice reflects our character and goals, and we are properly held accountable.
As far as I know, all Christians agree with these things. Everyone understands that we can't choose God or even respond to him unless he enables us, and everyone agrees that we make real choices for which we are accountable. The difference lies in how people account for the interaction of God's actions with us and the activity of our own wills.
Arminians say that God's activity with us and our own wills interact in a mysterious way such that God's grace is necessary, but our choices are still our own. This position is sometimes called synergism.
Reformed theology prefers to see God as in charge. Because he knows how anyone will respond in any situation, he knows when he moves someone whether or not it will result in their conversion. Thus we think God has an overall plan, which includes the salvation of every individual. Why does God's plan include certain people as saved and others as damned? That, we don't know. Calvin says it is for God's glory. Some people (particularly those who are horrified by Calvinism) think he means that God glories in damning people. I think what Calvin means is that while we can't know why some are saved and other aren't, when we finally come face to face with God and understand, we will see that his reasons glorify him, i.e. that his plan was meant for good, and has loving reasons behind it.
But remember, God's plan is simply a plan. People still make choices, for which they can be held accountable. He doesn't force them to do it. But he understands and is responsible for the larger pattern into which their choices fit, and they can't choose him unless he enables them to do so.
I am not convinced that Arminianism and Calvinism even contradict each other. I think an Arminian account of how people make decisions can be combined with a Calvinist account of how those decisions fit into God's larger plan.
But there's a lot of rhetorical posturing that you have to find your way through. For example, suppose God speaks to two people in their hearts, and he knows that one will end up saved and another will not. An Arminian would say that God has enabled both, but one chose to respond and the other did not. A Calvinist would say that God saved the one and left the other in his sins. God knew that his activity would end up with the first person being saved. His overall plan includes that. So it makes sense to say that he intended it. Thus, we say he intentionally saved the first person. Similarly, although he may have spoken to the second in the exact same way, he knew what the result would be, and for an omnipotent being we have to assume that he intended to do what he actually does. Hence we say that he left the second person in his sins.
Despite the very great difference in the Arminian and Calvinist accounts, there's a sense in which they don't contradict each other, but depend upon our choice of how to describe God's actions. Unfortunately the history between Arminians and Calvinists is such that very few of them will agree that the positions may not actually contradict each other. Of classical writers, John Wesley seems at times to have come close to this understanding.
But it's also worth looking at Luther. Luther had a strong commitment to maintaining mystery. He felt that there were things we couldn't understand, and that trying to penetrate things left dark by God would lead to trouble. While later Lutherans can at times be seen as synergistic (at least in how they deal with our situation after our initial conversion), Luther seems to have believed in election as strongly as Calvin. Like Calvin, he believed that we can only be saved when God graciously decides to do so. Thus Luther would agree with the Reformed tradition that God actively saves those who are saved. However he considered the understanding of why God allowed or even willed some not to be saved to be associated with God's "left hand." Today we might say it is part of the "dark side." He believed we don't have enough wisdom to understand God's ultimate reasons for this, and that we should leave it as a mystery. We should concentrate on what God has revealed, which is that he will save anyone who has faith in him, and that this faith is a gift from him.
You may recognize my first response as being somewhat Lutheran. Current Reformed and Lutheran bodies have largely stopped fighting about the (largely minor) differences between our two traditions. The PCUSA and ELCA (the largest Lutheran body) are now in full communion. Thus I think PCUSA folk are free to be guided by Luther at times.
