If you look at your question from the risk-reward standpoint, the answer is clear. There is no risk to writing this letter, and it is not a costly or time-consuming activity. The worst thing that can happen is that you will not get the job that was not offered to you in the first place.
Although the most likely outcome is no gain at all, there is a small chance that such a letter can help you. Sometimes the candidate who is the first choice does not jump in and demonstrate choice behaviors, and that can spell an early departure. There are other cases where such a candidate finds that this dream job is a nightmare, so he or she rushes the exit.
In such cases, employers often return to the original candidates for the position. When your file is reopened and your follow-up letter is sitting right there, it can only do you good. The letter obviously will not guarantee a job or a callback, but if it increases the likelihood of these outcomes by just one percent, why give that up?
A follow-up letter is a good thing, and it can help set the stage for good things to follow.
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