Does a letter from a lawyer work to resolve disputes? Not usually. And having opposing lawyers write letters back and forth is a great way to incur legal fees, but not so great in getting anything resolved.
The inherent problem with letters is that there is no penalty for ignoring them. A lawyer can threaten that something is going to happen if the letter is ignored, but that’s an empty threat. At times I have seen lawyers engaged in letter-writing campaigns that last for years—incurring tens of thousands of dollars worth of fees, but accomplishing nothing. Simply put, a letter will never force anyone to do anything.
Filing in Court, however, is a different matter altogether. Once an action is filed in Court, now you no longer are making empty threats. And a resolution will be obtained one way or the other once a legal action is filed because ultimately the Court will make a decision on the case at time of trial. Once your legal action is filed, you can talk compromise (if you want to) because if you are ignored, then you have the Court process to fall back on. In the words of Teddy Roosevelt: you should talk softly and carry a big stick!
Personally, I have a one-letter rule on pre-litigation matters (unless, of course, a client instructs me to do otherwise). I strongly caution against getting into a letter-writing war. They usually last too long, accomplish too little, and cost way too much.