I'm quite certain that God does speak to our hearts.
Yes, I think He does, too.
It is like the thoughts in our minds, but it has a quality that sets it apart. It is not a "voice" as such, but an impression in our spirit - because the Holy Spirit is a spirit and communicates with us spirit to spirit. What happens is that we interpret what has been received in our spirit with our mind. It is like, "The Holy Spirit is saying something to me, and I think it is...etc." But in order to know whether it is the voice of the Holy Spirit or our own thoughts, we need to be totally familiar with the written Word, and be people of prayer.
So, what does the Bible tell me the Holy Spirit will do in my mind and heart? Scripture tells me that the Spirit convicts me (John 16:8), illuminates my mind to God's truth (John 16:13), puts to death the deeds of my flesh (Romans 8:13), works in me both the ability and desire to do God's will (Philippians 2:13), produces the "peaceable fruit of righteousness" (love, joy, peace, etc.) in me (Galatians 5:22-23), leads me (Romans 8:14), and bears witness with my spirit that I am God's child (Romans 8:16).
Does the performance of these things in my life by the Spirit require Him "speaking" to me in the way you've described above? I certainly think He acts upon my mind and heart, conforming me to Christ in my thought-life and desires. I'm not sure this is always or even mostly a transformation of which I am conscious, however. In any case, I don't expect or look for leading or instruction from God in the form of inner impressions or "voices." I have His word that Paul the apostle has written is entirely sufficient to communicate to me God's truth and will for me (2 Timothy 3:16-17)
It strikes me as...odd when a believer tells me he has an inner leading from God (whatever that may mean) but then discerns the truth of it by resorting to Scripture. Why not simply operate from Scripture all the time? Why this desire to "hear God's voice" when it is already speaking in the pages of the Bible? Forget divine voices in my head; I'll just do what I know God has told me in His word to do.
The Holy Spirit told me one, "If I hear your voice in prayer a lot, then you will hear My voice to you." I find that He mostly speaks to me by directing me to a verse of Scripture that I have never memorised.
Where in Scripture do you read what the Holy Spirit told you about prayer and hearing His voice? I can think of no place where such a rule is communicated.
I memorize Scripture because God tells me in His word to do so (Psalms 119:11; Proverbs 3:3), because I always need it ready at hand in the daily fight I'm in against the World, my own flesh and the devil (Ephesians 6:17b; Matthew 4:1-4), and because it nourishes my soul (Jeremiah 15:16; Psalms 19:7-11; 1 Peter 2:2) and lights my way (Psalms 119:105). In light of all Scripture can do - and does - for me, I see little use for a divine "voice in my head."
I do, though, believe God brings certain verses or passages to my mind at times when I have needed Him to do so. In fact, I believe that if God speaks to my mind in any way, it is in this way - which is another reason why a thorough knowledge of His word is absolutely vital to my walk with Him.
I tend to have extended discussions with God about issues and questions, and while I am actually discussing them, new thoughts will come to mind that answer my questions! I see this as the Holy Spirit joining in the discussion, making it a two-way discussion where I end up having my questions answered.
I agree that God may guide one's thoughts, ordering them along a line that brings them to His truth. But, again, this is, in my experience anyway, a subtle, natural thing that doesn't involve a "voice in my head."
When I am wanting answers from God about any issue, I don't wait for a voice, word of knowledge, vision or a prophecy. I have a discussion with God about it, put my case, and then use my own judgment to find the way forward. I find that if He wants to guide me in a different direction, He gives me Scriptures that adjust my direction as if to say, "This is the way I want you to go."
Amen.