The first question needs to be "Are any ARES groups planning to use the Internet for emergency communications?". My answer would be 'none that I know of'.
Now, to backpedal a bit, there are people using WINLINK. WINLINK can operate without the Internet, if the correct pieces are in place, to do local routing of emails. But, most WINLINK setups I've read about are designed to direct the email traffic outside a disaster area to servers located where the Internet is still working. There are multiple, redundant servers that can be used for that purpose.
Still, that is infrastructure, and while it's nice to have some of that available, the most critical emergency communications will be needed in the first hours of a disaster, and will likely be tactical in nature. Voice is probably the preferred mode, but there are situations that definitely call for data comms, too, and WINLINK offers some major advantages for that. It's not encrypted but it's not man-readable, either, so it does have some security advantages over other digital tools that we have.
You need to use the BEST tool you have available. If all that's needed is a short hop to the nearest working Internet connection, it's nice to be able to use it.
A way to separate these things, but have them available for disasters is to use WINLINK for NTS and use more tactical things for ARES, but have ARES use NTS for traffic handling outside the disaster area. ARES folks should know how to use NTS, and vice-versa. And, they should know how to pass traffic by voice or CW if WINLINK doesn't work.
Around here, we're using voice and NBEMS for ARES, and working on getting WINLINK up and running for NTS.