If the Levites and Priests already knew the criteria for recognizing the true prophet and false prophet (Deut. 13:1-5; 18:21-22), then why did false prophets flourish among Israelites? Why is there no mention of the execution of false prophets by the hands of Levites and Priests?
The answer to this question is really quite simple. The reason that there were false prophets who “flourished” is because human beings are sinful. We always want to hear from teachers who tell us what we want to hear. The flesh seeks to hear from those who support our sinful nature and do not call us to repentance. All societies and all peoples have false prophets who tell them what they want to hear. This was a problem in Israel for sure, as Jeremiah pointed out so many times. Jeremiah 6:14 and 8:11 is a good example of this.
There are an unlimited number of things not mentioned in the Old Testament. In fact, there are relatively few examples mentioned of the enforcement of the laws given at Sinai. This does not mean that the laws were not enforced, but that God chose to tell us about other things in the Scripture. I am sure God knows what is most necessary to tell us in the Scripture, but he had to keep the length within limits, so there are many details we do not know about. However, unlike what you said, there is at least one example if the Old Testament of false prophets being executed. It is in 1 Kings 18:30-39.