I've used mostly webbing slings on Service Rifles over the past 40 years, but I have been using a lot of Leather slings the past 2 years. The webbing slings are infinitely adjustable, are cheap, and they work just fine in the rain.
Leather is a pain in the neck when new, does not do well in rain, but you can put together a 1907 sling a bunch of different ways, and one of the ways I have only been shown recently works great (in dry weather).
Turner makes a decent leather sling (I have several) and they make an all-weather one:
https://turnersling.com
Les Tam makes what many people say is the best, but you wait forever:
http://www.lestamslings.com/products.htm
Ron Brown makes a really good sling (I only have one of his, and I'm really impressed):
https://www.creedmoorsports.com/product ... department
Another decent sling maker is Eric Hollis (I just got one of his):
https://erichollisllc.com
A friend from Arizona named Lee McKinney makes good slings (I have several) but all I have is an eMail address for him.