Speaking in tongues in our private prayer life is for our edification. Even Paul prayed in the Spirit - For if I pray in an unknown tongue, my spirit prayeth, but my understanding is unfruitful. What is it then? I will pray with the spirit, and I will pray with the understanding also: I will sing with the spirit, and I will sing with the understanding also. (1 Corinthians 14:14,15)
But the reference you had used proved that tongues cannot be a prayer language because it is not fruitful as being edifying to oneself unless someone interpret it which is what Paul was praying for to have that understanding.
1 Corinthians 14:
13Wherefore let him that speaketh in an unknown tongue pray that he may interpret.
14For if I pray in an unknown tongue, my spirit prayeth, but my understanding is unfruitful.
15What is it then? I will pray with the spirit, and I will pray with the understanding also: I will sing with the spirit, and I will sing with the understanding also.
The verses above are not talking about speaking in tongues. This was not available until the day of Pentecost. The above verses are talking about God giving you words to speak when witnessing to others. But beware of men: for they will deliver you up to the councils, and they will scourge you in their synagogues; And ye shall be brought before governors and kings for my sake, for a testimony against them and the Gentiles. But when they deliver you up, take no thought how or what ye shall speak: for it shall be given you in that same hour what ye shall speak. In other words - God's got your back - He will tell you what to say via the Spirit within you.
The point of the reference was signifying how the Holy Spirit speaks: as in what He hears: that He speaks, and so when we speak whethor it be our native tongue which was teh referene in Matthew 10th chapter was for or by the gift of tongue in other men's lips as pertaining to the gift at and after Pentecost: the Holy Spirit is speaking as per whatever words He hears to speak:
So then tongues as being used as a prayer language is redundant if the Holy Spirit speaks what He hears from God.
Tongues that are interpreted are for the edification of the church which Paul admonished the Corinthians for not doing. Speaking in tongues is a manifestation of the Spirit within. I seek to please God and if He tells me through His word that speaking in tongues is for my edification; then I will pray with the spirit so that I may be edified.
Tongues that are not interpreted would only be for other men of foreign tongue to hear and to understand: they may not believe what they hear: but that is what tongues are for: tongues that cannot be interpreted and rationalized as being a prayer language is not scripturally supported.
Paul spoke for what purpose the manifestation of the Spirit was for: to profit the body withal: and tongues as listed is followed by the interpretation of that tongue: tongues was never meant to be a stand alone gift: that was why Paul exhorted prophesy over all spiritual gift because it stands alone to edify as a gift.
1 Corinthians 12:
7But the manifestation of the Spirit is given to every man to profit withal.
8For to one is given by the Spirit the word of wisdom; to another the word of knowledge by the same Spirit;
9To another faith by the same Spirit; to another the gifts of healing by the same Spirit;
10To another the working of miracles; to another prophecy; to another discerning of spirits; to another divers kinds of tongues; to another the interpretation of tongues:
11But all these worketh that one and the selfsame Spirit, dividing to every man severally as he will.
12For as the body is one, and hath many members, and all the members of that one body, being many, are one body: so also is Christ.
In context: the admistration of the manifestations are to profit the body of Christ: so when Paul says when he speaks in tongues as edifying himself: he expounds further that when so doing: it is unfruitful even to himself which is why he prays that someone may interpret.
Thus tongues as a prayer lnaguage does not edify oneself and thus tongues cannot be used as a prayer langauge as it must comes with interpretation to even be understood and be fruitful to the one speaking in tongues.
Yes, we are to study to show ourselves approved rightly dividing the word of truth. Vain babblings - empty sounds, sounds containing nothing - I wouldn't put speaking in tongues in this category.
I would put using tongues as a prayer language in that category. It is vain.
I would rather pray in my language, knowing what I had prayed for so that when Jesus answers my prayer, I can give thanks and glory to Him in His name.
John 14:
13And whatsoever ye shall ask in my name, that will I do, that the Father may be glorified in the Son.
14If ye shall ask any thing in my name, I will do it.