While even thin leather straps are unlikely to break, they do tend to stretch (especially in wet weather). You need the proper stuff they use for harnesses, saddles and holsters. A good, thick strap might be very stiff at first, but once it breaks in, it will last for life. You do also get very supple thick leather from places like Argentina, but that pushes up the price a lot. Some leather straps use suede on the inside (very soft and supple and nice to wear) and some really stiff stuff on the outside for durability and stiffness.
Speak to Hammeron - she's the one who made the straps above. While she may not respond on forum, you can PM her:
http://www.guitarforum.co.za/pm/?sa=send;u=3309