What is the best way to aggregate news headlines?

Skimfeed will give you a firehose of headlines.

It's usually enough to make me realize that I don't want my news headlines aggregated, and then I go back to curated collections and serendipity.

Twitter. Just follow all the newspapers/media outlets you’re interested in.

The best way is to aggregate them to /dev/null.

I am only half-joking.

Any open-source projects or closed source with API?

For what? Giving us more information will help us answer your question. If you want a way to collect news from different sources into one place then there are a lot of news aggregators sites and apps, here are some examples you can try: https://feedly.com/ - https://app.usepanda.com/