I rarely have to dry clean the dress shirts under normal wear conditions. I'm guessing it depends on the GSM of the shirt.
I don't know if I would agree dry leaning defeats the purpose. Most cities have a dry cleaner and dropping it a dry cleaner is going to be faster than washing, drying, and ironing it yourself. At least it is for me. Plus, you would need to find the right kind of detergent without enzymes or whatever while traveling.
It's still perfectly functional. There's just some pilling and lack of color homogeneity and certainly doesn't look new. I'll still wear it so long as I don't really need to be dressed up which I rarely am these days. (Like almost never.)
You're right that it wouldn't need to be dry cleaned after each use under minimally sweaty conditions. That said, while I have occasionally had laundry done in the middle of an extended trip, by and large I don't want to be dealing with laundry and dry cleaning if I don't need to.
Yeah, I bought one and it didn't really hold up very well. Having to dry clean sort of defeats the purpose of having a travel shirt.
In general, I really like merino wool but it's expensive and fairly fragile even if I avoid putting it in a dryer.