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Hi, great question; yes, that would be doable through the API, but we're still putting the final touches on that.
The "full API" would necessarily expose the entire complex model that's used behind the scenes (e.g. differentiating between jobs, job templates, and executions, etc), which from our experience is a real pain to work with, because it requires you to become experts of the domain model. Then you have to authenticate to it, etc.
So, perhaps we'll also have a more intuitive API that looks a lot more like the web interface, and exactly like you said, has a simple url for a "kick me now" button of sorts.
In the mean time, i'd definitely check it out to see how close it is to your requirements, and we would be happy to work with you to get a PoC going. It's still 1.0, but that's how we improve and get to 2.0 and so forth :)