Long URLs are hard to remember, look ugly and sometimes are simply too long to fit in the field. That's when you have to use an URL shortener. These free to use online services are very often on very short domain names, so the complete URL might be even shorter than your domain name only. The final shorter link will not be only visually more appealing, easier to remember, but quite often you'll also get a click tracking stats.
And yes, the goo.gl shortener is discontinued (I do get this question occasionally via email). Google announced that the links will continue to work, but you can no longer create new links. But no worries, there are many options to choose from.
And if you want a real domain name, check Namesilo - great domain prices, .com domain for only $8.99!