My opinion--yours may vary. Most of below comes from knowledge of 2011 through 2016—recent years may have changed some of this, although I haven’t noticed much as I look at recent years documentation.
The Ford Super Duty brochure showed that the F-250 used a lighter capacity axle from 2011 through about 2015. There was a great debate online with most contending that the same axle is used in both the 250 and 350 SRW for mass production cost saving. It is possible that the lighter axle was used for 2011 Job 1 250s to use up existing supplies from 2010. However, everyone now believes that the axles are the same from at least 2012 and possibly 2011. The F-250 is typically derated to 9900 or 10,000 lb. GVWR even though it can physically handle more. This allows insurance and licensing cost savings in some states. The F-350 since 2011 adds the auxiliary spring and that’s about it other than labeling. So, yes, in agreement with most of above, tires are likely to be the limiting factor. I have noticed in recent years that the 20” tires can get as much as 14,400 GVWR.