“So do the elect have any choice in the matter?”
Strictly, no, we have not chosen Christ:
Joh_15:16 Ye have not chosen me, but I have chosen you, and ordained you, that ye should go and bring forth fruit, and that your fruit should remain: that whatsoever ye shall ask of the Father in my name, he may give it you.
On the the other hand, there is a sense that all those who are the elect, willingly come to God, and not by force, as Calvin explains it in his commentary on John 6:
“To come to Christ being here used metaphorically for believing, the Evangelist, in order to carry out the metaphor in the apposite clause, says that those persons are drawn whose understandings God enlightens, and whose hearts he bends and forms to the obedience of Christ. The statement amounts to this, that we ought not to wonder if many refuse to embrace the Gospel; because no man will ever of himself be able to come to Christ, but God must first approach him by his Spirit; and hence it follows that all are not drawn, but that God bestows this grace on those whom he has elected. True, indeed, as to the kind of drawing, it is not violent, so as to compel men by external force; but still it is a powerful impulse of the Holy Spirit, which makes men willing who formerly were unwilling and reluctant. It is a false and profane assertion, therefore, that none are drawn but those who are willing to be drawn, [153] as if man made himself obedient to God by his own efforts; for the willingness with which men follow God is what they already have from himself, who has formed their hearts to obey him.”
Thus our willingness, in other words, is from God who made us willing. So that we choose to believe in Christ, yet it is only because Christ first chose us.
Always seeming paradoxes.
The only logical way to grasp this concept once the premise is accepted is to throw out a sequential flow of time. Once again we must leave our way of seeing time and the universe behind.
So we are again being forced to accept an un-provable concept and wrap a theory around it to make it work.