I’m one who thinks that Matthew was written before Paul’s letters for various reasons.
I think one clue is that Paul asks his readers to remember 'the day of the Lord will come like a thief in the night.
Taken in isolation this would by very strange. What is the ‘Day of the Lord’ and why of all things would he compare it to ‘a thief in the night’?
bible.cc/1_thessalonians/5-4.htm
Paul doesn’t tell us. If we read Matthew though we can see where Paul gets his words from.
Matthew, especially 24:42-44 but within the context of the whole chapter.
realdouayrheims.com/files/matt24.pdf
Also if we look at Paul arguing that Gentiles should be exempt from the Law, on many occasions he looks like he is responding to Matthew’s Gospel. At one stage Paul writes rhetorically ’ What are we saying then, that faith has replaced the law?’ He then goes on to say no, but really if you read him in full he is pretty close to saying that.
bible.cc/romans/3-31.htm
This seems to be in response to Matthew’s writing saying the law has not been abolished.
bible.cc/matthew/5-17.htm
Matthew also gives account of people being put right by their actions, faith if you like. Paul does not miss this opportunity to argue that faith is a justification amongst the gentiles and not Jewish law.
In fact if you read Paul’s letter to the Romans and count how many rhetorical questions Paul asks there it seems that he is certainly arguing against someone but in a very respectful and indirect manner. I think its Matthew’s gospel.
It is hard to see how Matthew could be written after Paul leaving open the ammunition Paul uses in his writings.
It is easier for me to see Paul trying to delicately tread around the authoritative writings of Matthew (and the other inner circle apostles) is arguing for his main thesis.