To get back more specifically to the OP, Paul asks 'Do all speak with tongues?' Literally, that would be rendered 'not all speak in tongues speak'. (I Corinthians 12:30.)
There are those who say stuff like, "He's talking about giving a message in tongues in the congregation, not tongues as a prayer language' or 'not tongues as initial evidence.' Some Pentecostals treat 'initial evidence' tongues as the same thing as tongues as a 'prayer language' and some do not.
But verses from which we get the whole idea about praying in tongues in the first place treat speaking in tongues in the congregation as the same kind of thing as speaking in tongues. For example from I Corinthians 14,
4 He that speaketh in an unknown tongue edifieth himself; but he that prophesieth edifieth the church.
5 I would that ye all spake with tongues but rather that ye prophesied: for greater is he that prophesieth than he that speaketh with tongues, except he interpret, that the church may receive edifying.
Here we see that one speaking in tongues edifies himself. But Paul treats that as the same thing as the kind of tongue that is interpreted in verse 5.
Or what about these verses:
27 If any man speak in an unknown tongue, let it be by two, or at the most by three, and that by course; and let one interpret.
28 But if there be no interpreter, let him keep silence in the church; and let him speak to himself, and to God.
So what is the evidence that everyone can speak in tongues? In Acts 2, it says of the disciples that 'they began to speak with tongues as the Spirit gave them utterance.' A more natural reading is that this refers to the 120. Some insist it refers only to the 12 apostles. Be that as it may, in Acts 10, when the Spirit fell on the Gentiles in Cornelius house.
Acts 10
45 And they of the circumcision which believed were astonished, as many as came with Peter, because that on the Gentiles also was poured out the gift of the Holy Ghost.
46 For they heard them speak with tongues, and magnify God. Then answered Peter,
47 Can any man forbid water, that these should not be baptized, which have received the Holy Ghost as well as we?
48 And he commanded them to be baptized in the name of the Lord. Then prayed they him to tarry certain days.
Does 'they heard them speak with tongues, and magnify God' prove that each individual there spoke with tongues? That seems rather inconclusive. In one Gospel, the disciples ask why the ointment wasn't sold and given to the poor. In John, we discover it was Judas Iscariot who said it. Sometimes when 'they' do something in the Bible, a representative of a group do it, or some in a group do it. Also, if half the people there magnified God in Greek or Aramaic and the other half spoke with tongues, wouldn't the wording of the passage be true.
There is also this passage in Acts 19,
2 He said unto them, Have ye received the Holy Ghost since ye believed? And they said unto him, We have not so much as heard whether there be any Holy Ghost.
3 And he said unto them, Unto what then were ye baptized? And they said, Unto John's baptism.
4 Then said Paul, John verily baptized with the baptism of repentance, saying unto the people, that they should believe on him which should come after him, that is, on Christ Jesus.
5 When they heard this, they were baptized in the name of the Lord Jesus.
6 And when Paul had laid his hands upon them, the Holy Ghost came on them; and they spake with tongues, and prophesied.
So if some of them spoke in tongues and prophesied, or half spoke in tongues and half prophesied, wouldn't these verses fit that situation?
Experientially, a lot of people do experience speaking in tongues along with an experience of Spirit-empowerment. But some Pentecostal churches that emphasize speaking in tongues as 'initial evidence' have those people who keep coming up at altar calls, hoping to speak in tongues, for years. I have also seen, mainly Charismatics, trying to coax or almost coach some sort of utterance out of people to get them to make some noise as if that is going to necessarily be speaking in tongues. Not all do that, but I've seen that sort of thing. I don't believe the Holy Spirit needs that kind of 'help.'
The initial evidence doctrine brought a bit of division with it. It split the Pentecostal Holiness church. FF Bosworth left the Assemblies of God over it and went back to the CMA and had a great healing ministry after that from what I've read. He wrote a tract, "Do all speak with tongues?' about the issue back then.
Experientially, I know a brother who goes to an A/G and interprets tongues. There is another interpreter, and the other guy sometimes beats him to the punch and gives the same interpretation he was going to. (That sort of thing happens, where someone else gets the same interpretation of tongues, which is pretty cool.) But this man has never spoken in tongues himself. He's prophesied and interpreted tongues.
I think we have to be careful not to make up really clear doctrines where the Bible doesn't, and not people into bondage into our own traditions and mindset. We shouldn't promise God will work in a very specific way when the Bible doesn't not clearly teach that He must.
I see a better case in scripture that anyone in the church could potentially prophesy than for everyone being able to speak with tongues. If you look up the word 'dunast'e' in I Corinthians 14:31, it means 'to be able to'. "For ye may all prophesy" it says in the KJV. It makes sense that 'ye' means the congregation.