SS barrels don't melt. But when subjected to high heat, they warp much more easily than chrome-moly. So you have to sight SS barrels in again after every firefight -- and soon they just won't hold a sight-in anymore.
It is not a hardness issue. The problem with SS is on a molecular level -- the same properties that allow stainless to be more resistant to corrosion also allow it to warp more easily.
Weapons designers have been trying to find a type of SS that will work with Big Guns for decades, and they still use stellite lining instead.
As far as firing two 30-round mags back-to-back in semi, with all due respect, it's almost impossible to duplicate the heat-generating effects of FA fire in semi. I often fire three 30-round mags back to back in FA -- that's about all I will do for fear of damage -- and that 90 rounds in 15-20 seconds will have the gas tube starting to glow. Touch the barrel area and you'll get a nasty burn -- I still have the scars. And three fast mag dumps are not uncommon in a firefight.
I have also witnessed (not on my gun, thank you!) folks dumping three or four Betamags back-to-back in a '16, and you can actually get the barrel behind the FH glowing (not smoking, I mean glowing red) before the gas tube melts. Now, 300-400 rounds in a minute or two is excessive .... but if my bacon was in the fire, I'd like to know I could do it.
When you are subjecting a barrel to that sort of stress, you want it made out of the metal which has the highest chance of coming out in a still-operable condition.
Beyond the SS-versus-Chrome-moly debate, the milspec for chrome lining is to promote easier extraction under FA fire and heat; to reduce throat erosion under the same conditions; and to increase the bore's resistance to pitting and rust in humid conditions.
With today's technology, though, it's impossible to build a chrome-lined barrel to the same tolerances -- and thus accuracy -- as a match unlined barrel. So the SPR makers are sacrificing some of their margin of durability as a trade for increased accuracy. From the reports I've seen, they are happy with the trade-off.