More than likely, when those businesses hire English grads, they do so on part-time contract basis.  Make no mistake, grads: this means you are successful!

That means the successful degree-holders require a skill not taught in universities: how to sell.  How to understand what potential clients REALLY want. How to present themselves as valued company assets who understand this, and--above all--deliver on it. How to get another contract before the one you've got is complete.  How to extend that one-contract assignment into a continuing contract that spans YEARS of work. See this stuff in university course offerings lately?

One reason I believe there are many under-employed or unemployed degree-holders: they don't understand and appreciate sales. They've likely never learned what they really need to learn.