It's pretty common for a doctor's office or lab to tell you "don't call us, we'll only call you if the results require a follow up, otherwise you're fine". I hate it, and I think it's a terrible practice, but it's one I see a lot in medical related settings.
Here's a snippet from my region's Covid-19 vaccination pre-registration page:
Note "We ask that you do not call to check your status" - Don't call them, they'll call you.
Why I hate it
Things go wrong all the time. Computers crash, people mis-enter data, things get misplaced, user interfaces confuse people, buildings catch on fire, people take sick days, etc. etc. etc. While steps can (and should!) be taken to reduce these issues, they exist.
By telling people not to follow up they're removing the only check from someone incentivized to make sure it gets done. If I call the lab and there's no results from my test a week later they'll realize something got misplaced and bring me back in. If I call into the Covid folks and they can't find my registration we'll discover that it never got submitted, and fix it.
Without that person following up there may be no opportunity for the error to be corrected.
... those calls are a drag on the system!
Yes they are. But so is not catching the thing that system was designed to catch. That lab test that didn't actually get run could have caught a problem when it was still fixable with medication, rather than surgury. That missed covid-19 registration could mean that the higher risk individual ends up in the ICU with covid-19.
... automation will fix it
How? Developers make mistakes all the time. Spam filters lose/hide mail. Data is mis-entered by humans. The UI those humans entered can be non-intuitive.