First, I agree that the example, as you stated it, does indeed sound a little strange. But I believe that it is the (over-)simplicity of the example -- a very short sentence, with no context around it -- that makes it sound strange. And I'm not saying anything about you; of course it is the case that these are precisely the kinds of examples that are used when teaching grammar.
In my scenario below, I add what I believe is the much needed context -- I include the original sentence as part of a larger sentence. When you read it out loud, hopefully you will agree that it sounds exactly as it should!
In this scenario, some townspeople are complaining to a councilman that they have not yet received an important letter about changes to one or more of the County's tax laws. So, the speaker (the councilman) wants to let everyone know that a process was started a few days ago, in which a letter detailing the changes will be sent to every household in town, but that the process has not yet finished. That is the important point: it is a process that was started in the past, and it is still ongoing. Not all letters have been sent.
Secondly, in order to even justify using the passive voice for this, we can assume that in this scenario, the councilman is a crafty politician who does not want to be held personally responsible if anything goes wrong, for example if a large number of families eventually complain that they never received their copy of the letter. So, he never wants to mention who will be sending out the letters. He has decided that he wants to the make the "Letters" the subject of his sentence when he promises everyone that they will receive their own personal copy of the letter. Therefore, he uses the passive voice.
Letters have been being sent out to a great number of the households in this town already. Some households have received their copy of the letter already. Just to be clear, these letters started being sent out last Friday, which was when this office first learned of the changes that the County Commissioners made to the tax law.
I fully acknowledge that not every household in town has received their own copy of the letter yet; there are indeed more letters to be sent and more households to reach; but the letters will continue being sent out until each and every family in this town has received their own copy of this very important letter that spells out the changes in the law.
Now that I have used the phrase in a larger context in which I make it clear that there is an on-going process which was initiated in the past but has not yet completed, I can even use your original phrase all on its own and it won't sound strange at all. But it will sound even better if I insert the word "already".
Let me repeat: Letters have been being sent already; in fact, three people approached me before this meeting started and they informed me that they had received their copy of the letter yesterday afternoon. So, please be patient.
If some of the letters have already been sent, and if others have yet to be sent, then he pretty much has to use present perfect progressive to accurately and honestly convey this fact. And in his attempt to cleverly avoid taking (at least) full responsibility in case anything goes wrong, he decides to make the "Letters" the subject of his sentences. Once he makes that decision, he also has to go with the passive voice.
But even if someone disagrees with me and says, "He doesn't absolutely have to go with the passive voice", that's fine. That's all the same to me. What I would say, however, is that once the "out-of-context" sentence "Letters have been being sent" is put into a larger, meaningful context, I personally have absolutely no problem with the way the phrase sounds.
In this final example, note the use of the original phrase, to give at least a hint (to leave the door open) that this is possibly still an on-going process; and then note the second phrase in bold, which leaves no doubt that the process has not yet finished. I threw in the third phrase in bold just to add a little future into the whole mix. :-)
Let me make sure that everyone understands what has been happening with this situation in the past few days. Starting last Friday, these important letters have been being sent out. And although not all of them have been sent out, they most certainly will be. In fact, at this juncture, it is expected that all the letters will have been sent out by Thursday of this week.