Consider how often we might be doing something in our own homes and arrive at a universal design solution - intuitively - that we can introduce to our clients. The reason it is described as intuitive is because a solution to an issue wasn't being sought, or even considered. We possibly didn't even realize it was an issue - until we discovered it, along with the solution.
Take for example a grab bar located near the entrance to the tub or shower. We did not have one in our own home. Maybe we had never thought about the need for it. One day we slipped (but fortunately regained our balance without falling) getting in or out of the tub or shower and reflexively put our hand out to catch ourselves. Only there was nothing there to grab onto except the flat wall surface or possibly the towel bar. It suddenly occurred to us that a small (12"-15" or so) vertical grab bar would be great to have - for those times when we slipped, felt dizzy, couldn't put all of our weight on our foot or ankle for some reason, had a muscle strain or cramp, or were recovering from an injury or surgery. This made sense then and it continues to be a good choice for anyone. For those people who are opposed to the term grab bar, there are assist bars, safety bars, and safety assists - all meaning the same thing. Not only that, but they come in a variety of colors, styles, and finished to allay the argument of the institutional look.
How about the time when we had our hands full of groceries, vegetables taken in from the garden, the mail, takeout food for the family, or any of many other possibilities, and two things occurred to us - both of which are universal design treatments. First, we noted that it sure would be great to have a place to set down what we were holding to free up our hands to open the door without balancing our load, juggling it, or attempting not to drop any of it. An entry shelf or entrance station does the job - a shelf, table, chair, bench, cabinet, or other piece of outdoor furniture.
Second, we found that opening the door by pushing down on a lever handle would be preferable to grasping and turning a round knob or trying to use a thumb latch that is nearly impossible to operate with just one hand (people have too small of a span between their thumb and fingers or don't have sufficient strength in their thumb to use this with just one hand). Additionally, if we were holding something in our hands, we could push down on the lever handle with the side of our hand, wrist, forearm, elbow, closed fist, or other ways. Without coming to an issue with a solution, the way to approach the issue became clear to use from experience - it was intuitive.
There are many more examples such as these where we discover a great solution that fits into the decor of a home without causing undue attention to itself, that works for most people (regardless of their age, physical size, or ability), and that possibly we hadn't considered before determining a fix to a situation we happened upon.
There are many design treatments that can be achieved through universal design just by looking at what is not present, what could be desired in a space, and then creating or recommending it to our clients. It's intuitive because it just naturally follows as a solution to the deficiency we observe.