I think I released my first item back in 2013 or so, and I made the same decision you almost did - my script relied on my own API for certain features. The problem was, if my API went down (due to an unexpected outage or maintenance), those features stopped working, and it was very poor user experience.
Basically, your goal should be to reduce the number of remote services your script relies on. This means less chances for there to be outages that affect your item. It won’t affect your approval chances, it just makes your script more reliable.
This doesn’t mean you can’t set up your own API. You can absolutely have an API for things like checking for updates or verifying licenses. Just try to make sure that none of the main features in your script rely heavily on it. (And if you need to use an API, it’s better to use a service who probably has better servers than you)