It's always good to be cautious and you're certainly not the first to wonder or worry about what happens if Pi-hole dies and is the only DNS server available. Because without it, your Internet (as you know it) would stop working!
I'm not sure if you know this or not, but many routers actually use dnsmasq to provide DHCP and DNS services, which is really no different then us; we have some additional software built on top of it--including our Web interface--but the actual reliability of DNS resolution will be comparable to that of a home router since they are both using dnsmasq.
It may not happen with every router--it's just something we have seen, which is why we mention it.