Paul's use of tongues in the first part of the passage is, I agree, to focus on his point of prophecy to edify the church. That does not mean, however, the example of individual private tongues (apart from edifying the church and purposed for building up of the individual) was a fictitious illustration to Paul's mind. The fact that Paul's emphasis was on edification does not negate the fact that he used tongues in another sense for his argument.
He did not use tongues in another sense. Let's take a look at chapter 13:
1Co 13:1 Though I speak with the tongues of men and of angels, and have not charity, I am become as sounding brass, or a tinkling cymbal.
There can be no case made that we have some ability to speak the language of angels. Paul here was using hyperbole. The point is that unless you speak in a manner that is understood by others who are around you then it serves no purpose corporately to do so.
Same in chapter 14:
1Co 14:2 For he that speaketh in an unknown tongue speaketh not unto men, but unto God: for no man understandeth him; howbeit in the spirit he speaketh mysteries.
Paul was in fact being redundant. Notice however, that the spirit being spoken in is a small "s". That means it is in our spirit. There is no indication that this was a leading of the Spirit of God.
Also, Greek was a common language as English is today. Most everyone spoke Greek but most every one also had their home language. Since there were many languages in a single church and most people preferred to speak that when they testified in a corporate service they often would do so in their own language. The problem arises out of this that not everyone in the service could understand what was being said. Paul was correcting this and encouraging them to either speak what is common or have someone interpret in a common language so that all who was in attendance could understand. Paul was saying in those first couple of verses that when you speak in your home language and no one understands you then what you say is between you and God alone and that does not serve the congregation.
There is no indication that the tongues being employed in this was unknown to the entire world. The short analysis of these chapters is love each other enough to consider others, speak what is common and save your home language for evangelism.