They are wrong. They are completely and utterly wrong.
You can have a stammer and still interview well. I once knew a guy named Jack. He's one of the funniest and most charismatic people I've ever known, and he had severe strabismus (lazy eye). I'm not talking about a minor case either. One of his eyes would roll off, staring in to the distance as you were talking to him.
The eyes have it, as they say, and when you have a conversation with someone suffering from strabismus, it can be extremely uncomfortable and distracting. Jack knew this, but instead of being sheepish about his condition, he would address it head on. He had a clever quip that he used to simultaneously draw your attention to his condition, as well as communicate that he understands it can be distracting. It became immediately clear that his confidence out-shined his condition.
I know this isn't easy advice, but you have to learn to do the same. Based on the work in your Github profile and the posts on your blog, I'd say you're certainly worth more £15,000 a year.
Your employers are not the ones who believe you interview badly, you are! When you stop believing that and put yourself out there, you'll grow past it. Once you grow past it, you'll earn at a level appropriate for your skill level, rather than a social limitation.