Mr Hat's hat is not leather, but fur felt, a product made by felting rabbit fleece here in Australia for hats. I think all animal fleece can be felted, so it will depend a bit on where the had was made what animal fleece would have been used in Mr Hat's hat.
Felting produces a strong natural fabric that is largely impervious to water, so it is a good choice for hats. My experience is that fur felt hats give good protection from both sun and rain. Eventually, it is possible to soak a fur felt hat, but even then, it is still providing rain protection because the bulk of the water flows off the brim, unlike cotton hats.
I haven't used a leather hat, so don't know how well they might perform. They have always seemed much heavier than similar fur felt hats, and largely about style rather than function.
I rather suspect Indiana Jones' hat was a fur felt fedora, not a leather variant, but I haven't checked. I do recall that it was a modified version of a hat made by the famous London milliners, Herbert Johnson.