You're right. Sorry I meant the NS glue records. Those have a 1 day minimum TTL. If you're not already running multiple providers, switching after one is down won't help much.
Unless you use a better resolver than the standard glibc resolver on Linux (e.g. dnsmasq, bind or similar running locally and pointing resolv.conf at it), you appear doomed to slow lookups etc. if your first resolv.conf entry fails, as most of the resolv.conf options that might have helped (if you'd set them) simply don't work or doesn't do anything particularly useful in the versions used in the Linux distro's I've tested it on.
Only if you don't know what you're doing. The problem with DNS is that it might work even when it is misconfigured, and misconfiguration is the source of strange issues.
I think that we all have areas where we don't know what we're doing. This is one of mine. With all the talk of how obvious/important/easy it is to have a failover in place in case this happens, I'm having trouble finding a good resource about setting up a redundant DNS.
Running a droplet on Digital Ocean with Debian and Nginx.
Sounds like we're begging for someone to write a nice blog post for how they set up redundant DNS across multiple providers "the right way"... Sounds like it would hit the front page in short order if anyone is willing to share how they think this should be mitigated, and specifically how to expect common clients to behavior in that case when faced with the various types of outages that may occur!