I think the key distinction here is "run my new function through the entry point of the application" vs "run my new function from a test". You leave out whatever startup time is involved from the rest of the application to get to that function, & you have a tiny feedback loop to test the function.

You have your test which gets the response, and you can still log the output to see what it looks like. It's a little slower at first since you have to write the test, but I'm not sure I see a big distinction vs "have a small script that sends the request".