I agree. I see two manifestations of tongues in the New Testament. One is the manifestation of the gift of tongues with accompanying interpretation. The purpose of this is ministry in the assembly. Not everyone will be used in this gift. I believe that is what is being said by the rhetorical question, "Do all speak in tongues?"
But I also believe that any believer who has received the baptism of the Holy Spirit can pray in tongues. Paul makes this clear distinction when he says that he thanks God that he speaks in tongues more than any of the church at Corinth, but that he would rather speak in a known language in the assembly. Obviously if Paul is speaking in tongues when he is not in the assembly he is not exercising a ministry gift of tongues and interpretation since those gifts are specifically for ministry to the body. Scripture says that the gift of tongues is for the edification of the body. Yet is says that praying in tongues is for the edification of the person praying.
As to the initial evidence, I have an observation to add. First, I do NOT believe that a person MUST speak in tongues in order to have received the baptism of the Holy Spirit. I think that perspective goes beyond what scripture teaches. Yet I see that in most instances in scripture, speaking in tongues accompanied the receiving of the baptism of the Holy Spirit. I think there is a good reason for this.
I think the reason is that faith plays a huge role here. In order to pray or speak in a tongue unknown to you, you must exhibit faith. Anything that we receive from God, we will receive by faith. Receiving salvation, or receiving the baptism of the Holy Spirit both require a response of faith toward God. I think speaking in tongues is often, perhaps even usually, that response of faith that allows us to receive the gift that God has given us. If I am praying with a person to receive the baptism of the Holy Spirit, I will usually encourage them to take that step of faith.
I am not sure that there has been an over-emphasis on tongues, but I do think there has often been a wrong emphasis and even an odd or flaky emphasis in many cases. Many have focused on the manifestation to the exclusion of the purpose of the manifestation. It is much like focusing entirely upon saws and forgetting that there is wood to be cut.
According to 1 Cor. 2, we know the things of God by the Spirit of God, and being baptized in the Holy Spirit allows us to know these things in a greater way than we did before. When I do not know how to pray as I ought to pray with my intellect, I nevertheless know in my spirit because of the power of the Holy Spirit and the communion that my spirit has with His. This is why, I believe, that scripture makes it plain that when I pray in tongues my spirit prays, but my understanding is unfruitful. So when I do not know how to pray, I, by faith, pray in tongues, believing that I can be effective in prayer through the Holy Spirit. It is the purpose of tongues, not the tongues themselves, that should be the focus. When I am in need of edification, I get alone with God and pray in tongues, because scripture tells me that when a man prays in tongues he edifies himself. When I need wisdom and do not have it, I will pray in tongues over the situation.
I think tongues is an important part of the baptism of the Holy Spirit, else Paul would not have given it so much attention. But we must remember that tongues is not the point, but rather the tool that was given us to use in accomplishing the point. When our focus becomes misdirected, we get ourselves off track.