Personally I think it’s much easier to ask for those domain owners to create the txt record. (As it would save time and money) so you could automated the process.
(What I did is, switch a domain deal with all auth and report to an api-supported DNS provider, map all domains I need to control under this domian then do the automation work)
A public DNS server is only necessary if you hold lots of domains( since you would need to update it regularly and ensure speed and reliable, it’s a crazy thing if you just wants to hold required text records)