have an additional external DNS server
While I agree with you that additional DNS server is without a question a good thing, on this you need to understand that if you set up two nameservers on your laptop (or whatever) they don't have any preference. So, if you have a pihole as one nameserver and google on another you will occasionally see ads on things and your pihole gets overrided every now and then.
There's multiple ways of solving this, but people often seem to have a misinformed idea that the first item on your dns server list would be preferred and that is very much not the case.
Personally I'm running a pihole for my network on a VM and if that's down for a longer time then I'll just switch DNS servers from DHCP and reboot my access points (as family hardware is 99% on wifi) and the rest of the family has working internet while I'm working to bring rest of the infrastructure back on line, but that's just my scenario, yours will most likely be more or less different.
As far as I know it is the default way of handling multiple DNS servers. I'd guess that at least some of the firmware running around treats them as primary/secondary, but based on my (limited) understanding at least majority of linux/bsd based software uses one or the other more or less randomly without any preference. So, it's not always like that, but I'd say it's less comon to treat dns entries with any kind of preference instead of picking one out randomly.
But as there's a ton of various hardware/firmware around this of course isn't conclusive, for your spesific case you need to dig out pretty deep to get the actual answer in your situation.