Design can’t speak for itself any more than a tamale can take off its own husk. You’re presenting a solution to a business problem, and you’re presenting it as an advocate for the end users. The client needs to know that you’ve studied the problem, understood its complexities, and that you’re working from that understanding.
Stop trying to get your clients to “understand design” and instead show them that you understand what they hired you to do. Explain how the choices you’ve made lead to a successful project. This isn’t magic, it’s math. Show your work. Don’t hope someone “gets it,” and don’t blame them if they don’t—convince them.
Anyone who is the least bit connected to the design world needs to read Design is a Job. Too many people think design is solely the pixels on screen or the ink on paper, ignorant that design is all of the decisions and knowledge that lead to those pixels or ink.
Seriously, go read the book.