The answer to the first question, are ARES groups relying to heavily on the internet?
I'd say yes. DMR, Fusion, D-Star, all of that, to communicate any great distance relies on the internet to do the heavy lifting. Now with that being said, all of those digital voice modes will work just fine without the internet. You just can't communicate much beyond the range of your local repeater. If all you've got is an HT and you're relying on internet linked repeaters, you're going to not be terribly effective for very long.
Is that a problem? Maybe yes, and maybe no. The internet itself, the core routers and servers, isn't likely to go down, and if it ever does, we've probably got bigger problems to deal with. When people say "the internet is down", what's usually the case is that their local on-ramp to the internet is down, which to an end user has the same effect. In that scenario, using Winlink to route email outside of the affected area to a place where the local on-ramp isn't broken will certainly get the job done.
For preparedness purposes, as a former EC and DEC, I'd recommend that ARES groups should be practicing VHF simplex, HF NVIS operation, and providing (and regularly testing) backup power for their stations. I can talk to my county EOC (and a few adjacent counties) on 2 meter simplex (analog or digital voice). I can talk directly to the state EOC 130 miles away just fine on 40 or 75 meters using NVIS propagation, not to mention state EOCs in several adjacent states. And I can pass digital traffic to all those locations using those modes as well, with or without commercial power being available.
ARES members should also know how to program their radios by hand, without the use of a computer.
Any computer you intend to rely on for operating digital modes, should be kept isolated from any access to the internet.
Bottom line, ARES members should have multiple tools in their toolbox, and be trained to use the right tool for the right job, depending on the circumstances.
73 de N8AUC
Eric