Do you have am examples of "some websites" that you can post a link to? I wouldn't be surprised if certain specific details are left out (such as a sudden reason for the driveway failure or an expensive insurance policy)
My thoughts are:
A) It will depend on your specific policy. For example if I has a sudden failure of the roof, new water damage is covered 100% after deductible (up to the limits of the policy), but for a slow leak (that they think I should have known about) that has become rot there is a provision for $2.5k.
B) Do you really want to file a claim with your insurance? If they were to reassess your risk taking into account previously filed claims, how would you feel about a high premium or dropped coverage?
C) Some settling and cracking in a 33 year old asphalt driveway is pretty common (reason to deny). Have you been diligent in sealing cracks or seal coating the driveway as part of your household maintenance. (you could be blamed for making it worse than it otherwise would be, especially if it is covered), settling is often a sign of improper foundation/roadbed (does your insurance cover defects in installation that presumable happened before coverage began).
D) I would be surprised to see it covered by insurance, but I have seen things that should be routine maintenance paid for by insurance before. Years ago a neighbor of ours had some tree work partially paid for by his insurance, I am not sure it was a "claim" more of an associated benefit. Apparently the actuaries had done the math and found that partially paying for certain tree work (trees dying trees near the house or limbs overhanging the roof) could reduce expensive claims down the road.
E) How much is the repair/replacement? I'm sorry to say this, but I believe the only way to be sure that this will not happen again is likely going to be removing the current driveway, digging up under it, laying a proper bed under it (or in some other way addressing the issue, such as drainage and ditches to redirect water), and placing a new driveway. Asphalt moves too much to put concrete over it, re-topping the current driveway will just settle in the same shape as the current, and without fixing the underlying support issue the ground will shift again. If a quick Google search is to believed 33 years is a good run for an asphalt driveway (I don't want to believe since my driveway is already 28 years old).