Not everyone can speak in tongues, only those who are born again, and all who believe have this ability whether you know it or not. Paul wishes that they all would speak in tongues as they are under our control to speak or not to speak. That is the sign gift of Mark 16:16-18.
There is another gift of tongues that only a few have, and the difference is the direction. Mark 16 is a prayer language TO God. 1 Cor. 12 is an office. Those with the office of receiving messages FROM God that they know to speak in tongues do so for the profit of all in the congregation meeting. Interpretation of tongues is also an office, and they hear the tongues in their own personal language and speak it forth for everyone's benefit. However, even though these are offices, the Spirit can make anyone understand the interpretation at His discretion - like to the 3000 devout Jews listening on the Day of Pentecost HEARD all of them speaking in their own individual languages.
This is why 1 Corinthians 14:30 says "do all speak in tongues (office, messages FROM God), do all interpret (office). The answer is no. But all who believe receive the prayer and praise language of speaking in tongues of Mark 16.
Just to add to your enjoyable post.
If we carefully examine the context of why Paul asked the question "do all speak in tongues?" he is reminding us of what he said in 1 Corinthians 12 about the body of Christ being made up of many parts. Therefore his questions to ministry in the church. He could also ask, "Is everyone the pastor?' or "Is everyone a musician?" In this sense when he asks the question about tongues he is talking about the ministry gift of tongues in the church context.
But when he is saying, "I would you all spoke in tongues", and "I thank God I speak in tongues more than you all", he is not speaking of the ministry gift of tongues that is in tandem with interpretation of tongues. He is talking about personal prayer in tongues when he is alone with God. It makes sense, really, because why should he say, "I would that you all spoke with tongues" if he has already said that speaking with tongues in church is pointless because no one understands and therefore is not built up? And when he says, "I thank God I speak in tongues more than you all", if he spoke it in church more than all of them who were speaking in tongues in the meetings, wouldn't he be contradicting his own instructions?
Of course, you and I would see that quite clearly, but one who is prejudiced against tongues would not want to use basic comprehension because taking that sort of intelligent approach to what Paul is actually saying would cause his opposition to tongues go down like the Hindenburg - in flames! :-)
Oh! Consistency in comprehension, thou art a jewel!!