The critical service here is not the DHCP server, but the DNS server. If you are using Pi-Hole as your only DNS server (as you should be), if it goes down you have no DNS resolution.
The DHCP part of the network is only needed when a client needs an IP address. If you have the DHCP lease set for a week, for example, once the client gets its IP address it won’t ask for another until the lease period is nearing an end. So, if the DHCP function were to fail, your clients would continue working with the IP they have. However, since the DHCP function is on the same software that is providing DNS resolution, they would have a LAN IP but could not reach the internet with any domain they don’t have in cache.
Your router DHCP won’t take over since you should have it disabled so both devices aren’t serving conflicting IP addresses.
Pi’s are pretty reliable, but there are some things you can do to increase reliability:
Put it on a battery backup or UPS.
Put a clock module on the Pi - so if it loses power and then reboots when the power comes back, it will have accurate time. Pi’s don’t have internal clocks and don’t react well to sudden power loss and time keeping.
Use two Pi-Holes in parallel. If either fails, the other picks up the load.
Lastly, if you are routinely away from home; leave instructions for how to change the DNS in the router or setup something like TeamViewer where you can remote back to your network and fix the problem.