How can you see whether or not these shingles have physical damage? These shingles are 15+ year old 3-tabs that obviously don't have proper ventilation... but that doesn't matter in an insurance claim.
If she lives in a matching state, that roof is totaled. I guarantee that there is at least one tab on that roof that has either been lifted through the nail hole, creased or has undeniable mat transfer... all three of which ARE considered wind damage (would win appraisal 10/10 times) and covered by the homeowners policy.
Regardless, from what I can observe in the picture provided (which isn't much, but after observing it very closely, I am in full agreement with 'roofermann' in regards to believing there is exposed fiberglass), this 15+ YO roof with a 20-25 year useful life, worst case scenario, can easily be argued as non-repairable and as long as you are persistent and have the correct representation as stated above by AD, you will win that battle.
I would not hold off on replacing the roof given its age and condition. I especially would not give up on this claim, because at this point you have nothing to lose. They are already asking you to replace the entire roof on your own dime. Not sure what state you live in, but if the contractor has no luck on the re-inspection and worst comes to worst (It happens sometimes... we cannot argue policy and the insurance provider technically doesn't have to listen to us if we are acting as the General Contractor on the claim), I would definitely look into getting an appraiser to represent you on the claim before paying out of pocket. Even if you're contractor is not willing to cover the appraiser and potentially umpire fee's, it is still a much better option than the insurance provider is giving you now, which is 100% out of pocket. The appraisal process will end up taking some time, depending on the insurance company, maybe a very long time... but, as I said it beats coming out of pocket all day long.
If you by some crazy chance live around the Indianapolis area, I would be happy to take a look at it for you and devise a game plan if I can confirm any of the things I said above in regards to the picture (or find damage that the adjuster just so happened to 'overlook' while taking his one picture). Unfortunately, Indiana does not have a matching clause, which automatically voids the first and easiest option I proposed in this post.
I wish you the best of luck, truly!