Even better? Take your integration test, put it on a cronjob in your VPN/vpc, use real endpoints and make bespoke auth credentials + namespace, and now you have canaries. Canaries are IMHO God tier for whole system observability.
Then take your canary, clean it up, and now you have examples for documentation.
Unit tests are for me mostly testing domain+codomain of functions and adherence to business logic, but a good type system along with discipline for actually making schemas/POJOs etc instead of just tossing around maps strings and ints everywhere already accomplishes a lot of that (still absolutely needed though!)