If they were a close family member, probably, but if the person had been out of touch for years and wasn't on good terms when he left? I've known people who would just say, "Good riddance."
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Considering his wishes for anonymity, possible isolation during his life, I don’t think Lyle would be reaching out to a family member who would sum up his death as a “good riddance”, why to reach out to that person anyway? I have not had any contact with my parents and sibling for years and I can’t imagine reaching out to them on any occasion based on our relational history. I am not saying that Lyle had the same circumstances but it is unlikely that he would reach out for understanding or even apology if relationships had been strained/difficult for a long time.
I do also believe that idea of a suicide letter is a stretch. Like I mentioned earlier, it’s either a) Lyle didn’t have anyone b) have grown apart over the years and moved away – either was reported missing at some point but there is no record, or they never reported as they lost touch with him completely and it was normal considering their relationship, lifestyle, etc.
Also, if you are thinking that family could have gotten some benefits (financial) after his death, there are lots if legalities involved in all that and would have been reported missing first. Given that he has very distinct physical features and his face/body was intact after his death that would not be very difficult to accomplish. If they wanted insurance after his death, they would be looking to find his body/ locate him somehow to prove he’s dead.
Another reason that I don’t believe he wrote a letter is that he did everything to disguise his identity. I don’t think he was looking to hide his identity from anyone in particular, he wanted to hide from himself and everyone who
could identify him. I believe he grew up in an isolated/ conservative environment that had resources to promote growth and advancement but didn’t necessarily offer compassion, understanding or anything personal in a genuine way. Again, I don’t know if this is the case or not. Those are mere impressions that I hold.