Not All APIs Are Created Equal So Make Sure You Set Expectations Properly
I've heard of numerous API providers shutting down their API programs after a couple months because they didn't see the number of new users, and integrations they had hoped for. It is so easy for us technologists to shoot for the moon, while also simultaneously shooting ourselves in the foot when it comes to defining where the bar should be for our operations.
Remember, we are not all Twilio. For most of us our API will never be our primary product, and operating it will be less than a glorious affair--meaning you will never be written up in Techcrunch ;-). We are entering into the really mundane, business as usual phase of the API industry, and things will not be as exciting or sexy as they have been, and we need to make sure our APIs stay real, and reliable amidst all of the tech delusions we have on a regular basis.
Even if your API is not the next amazing tech sensation, it doesn't mean you can't operate following the lead of the giants like Twilio, Amazon, and Google. You can run a first class shop, even if you don't make the front page of Reddit for the launch, and get all the hacker kids building amazingly cool stuff in the first weeks. To do this you need to set some realistic goals for operations, that aren't based on the often fantasy world of Silicon Valley.
It is likely your company's website wasn't a major hit when it first launched either, and you didn't ditch it because it was underperforming. I'm guessing your goals were probably a little more realistic, and in alignment with your primary business objectives. Your API is no different than your web efforts, it will take time to do it properly, and probably require a few adjustments along the way--do not give up before you have a chance to find success.