This argument has never held water with me, for two reasons:
1) If it's true in the 21st century, it was true in the first century. But in the first century, not everyone had miraculous gifts. As Paul writes in Romans 12:
HCSB Rom 12:6 According to the grace given to us, we have different gifts: If prophecy, use it according to the standard of faith; 7 if service, in service; if teaching, in teaching; 8 if exhorting, in exhortation; giving, with generosity; leading, with diligence; showing mercy, with cheerfulness.In the first century, it did not make God a respecter of persons if one person was given a miraculous gift (prophecy) while another was given a non-miraculous gift (giving with generosity). Why then would the exact same arrangement in the 21st century make God a respecter of persons? Has God changed in the past 2000 years?
The text clearly states that God gives different gifts to different people, so whatever is meant by the phrase, "God is not a respecter of persons", it can not mean that God can't give one person a miraculous gift and another a non-miraculous gift. The text is black-and-white on this issue.
2) To make this argument, the phrase "God is not a respecter of persons" must be taken out of context. Here are the two relevant scriptures:
KJV Acts 10:34Then Peter opened his mouth, and said, Of a truth I perceive that God is no respecter of persons: 35But in every nation he that feareth him, and worketh righteousness, is accepted with him.and
KJV 1 Peter 1:17And if ye call on the Father, who without respect of persons judgeth according to every man's work, pass the time of your sojourning here in fear:In the Acts passage, the context is that God is not a respecter of persons in relation to who will be accepted with him.
In the 1 Peter passage, the context is that God is not a respecter of persons in relation to how a person's work will be judged.
But the argument requires a different, third, context: that God is not a respecter of persons in relation to who gets what gifts.
This third context is not in the text. The Bible simply does not say that God is not a respecter of persons in relation to who gets what gifts, and thus this doctrine is an addition to God's Word.
Whether God gives miraculous gifts today or not is an entirely different issue. But to use the argument that He does not do so because He is not a respecter of persons is to make an argument that is not in the Bible, and is actually opposed to a clear statement of the Bible.