In order to argue that God responds to our choosing him by then choosing us, one must presuppose that man is not only capable of doing so, but also that he would choose God even if he were capable. If we are not capable of choosing God, or if we would not choose God even if we had that ability, then God must first choose us rather than vice versa.
This brings us to two points the Bible makes about our ability to choose God: (1) Man does not have the ability to come to God; and (2) Man would not come to God even if he had that ability.
Let's talk about that first one for a moment. The Lord Jesus said, "No one is able to come to Me unless the Father who sent Me drags him [to me]" (John 6:44).
He says this in response to the Pharisees who do not believe (v. 36). He is now giving a response in John 6 as to why they don't believe. It is because the Father has not drawn them, and according to Jesus here, if the Father does not draw someone to Christ, "no one has the ability to come" to Him.
The word elkuō often describes a dragging of its object when it would either be against the will of the object (especially when the object is persons) or when the object doesn't have a will at all (e.g., like a sword). In fact, this is what it means in every unambiguous passage in the New Testament (John 18:10; 21:6, 11; Acts 16:19; 21:30; James 2:6). If this is true, God is not dragging men to Christ because He is responding to their choosing Him first. No dragging then would be needed.
Now, that brings us to the second idea that even if we had the ability to come to Him, we wouldn't. This, in fact, is the reason why we don't have the ability in the first place: because we don't have the ability to give ourselves a love for God that transcends our love for ourselves and our own rule over our own lives. We are always seeking to please ourselves instead, and so the Bible says that "no one seeks after God" (Rom 3:11). We don't seek Him because we are enslaved in love to our own sin/rebellion/self worship (6:20-23). Notice also in Romans 6:20-23 that we were freed from this enslavement, we did not free ourselves by choosing God. We were enslaved to its love and would not choose God when given the choice unless that enslaved love was first broken.
This is because before God saved us, we were dead in sin and needed Him to make us alive in Christ so that our desire would turn toward Him (Eph 2:1-10). Hence, we are not saved by any good work we did by choosing to follow God, but instead His choosing to free us from our self-love and raise us up with Christ.
So we love Him, not because we loved Him first, but because He loved us (1 John 4:10, 19). Our love and desire to choose Him is a result of His doing this for us first, not vice versa. And that is the way it has to be because of our enslavement to our self worship, and therefore, inability to free ourselves from that enslavement.
Hence, it was while we were still dead in our sins, enslaved to our own desires that rejected God as our Lord, that He decided to save us, predestining us from the foundation of the world to become His sons who will inherit all things. This decision was made, not on the basis of our choosing Him (as we were incapable and unwilling to do so), but on the basis of His love and mercy that He decided to give us. The reason for His choosing us then is found within Himself, His goodness and mercy and divine will, not on something that we would do like choosing Him (as we would not have done that).
Hence, we can say with Paul:
Blessed [be] the God and Father of our Lord Jesus Christ, who has blessed us with every spiritual blessing in the heavenly [places] in Christ, just as He chose us in Him before the foundation of the world, that we would be holy and blameless before Him. In love He predestined us to adoption as sons through Jesus Christ to Himself, according to the kind intention of His will, to the praise of the glory of His grace, which He freely bestowed on us in the Beloved. (Eph 1:3-7)
We can say this because only God is responsible for our salvation. He gave us the gift of faith (i.e., choosing Him in the first place--Eph 2:8-9; John 6:28-29; 2 Tim 2:25-26) and has saved us completely despite ourselves and our own choices that led us, and would have continued to lead us, far away from Him.
Finally, we could ask why God's predestination is worth anything if it only elects us to do the very thing we would have done had He not elected us in the first place. In other words, if God's choosing us is a response to our choosing Him, why does He even need to choose us? His election is worthless because He is merely electing us to do what we would have already done without His electing us. But I hope one can see why it is necessary for God to choose us first and then act to bring us to Himself, since we are incapable of freeing ourselves from the slavery of self-love and sin's mastery in order to love Him and choose to follow Him as Lord as a result of that love.
It's important then to understand that God's foreknowledge isn't talking about events where we would choose Him that God knew would happen in the future. In fact, in those verses that talk about our predestination, it's always people He foreknows in relationship with them, not actions and events. But the word there speaks of His eternal knowledge and will concerning these people, not God looking into the future as though He were gazing into a crystal ball to observe events that take place apart from His existence and actions.
So to put this all as simply as possible: We are too in love with our own self direction to ever follow God first. He has to free us from our enslaved love to our own self direction by making us alive in Christ, granting us repentance and faith in order to come to Him. We love and choose Him then as a result of His loving and choosing us first, and we could not be saved otherwise.
"You did not choose Me but I chose you, and appointed you that you would go and bear fruit, and [that] your fruit would remain, so that whatever you ask of the Father in My name He may give to you. (John 15:16).