You're never going to hear how a client is judging you based on your dress. Appearance is a subtle, unspoken factor. The client won't come out and say, "I decided you're not worth $50/hour," or "I don't think I'll do my next rating with you," based strictly on the way you dress. They'll just quietly stop coming, or never start to begin with.
People make decisions based on emotion, not logic. The logic has to be there, but emotion is what ultimately drives them. Dressing up a bit builds the "perception of value" in their mind. It makes them think, "This guy is serious about what he does. He's more mature than the guy at the other flight school. I bet he's more careful..." and on and on. It makes them feel good about flying with you.
The way you dress is sending out an unspoken message in the same way showing up on time, not using foul language, and having good hygiene sends out messages. It can go to both ends of the spectrum, too...if you show up in a full airline pilot uniform, under some circumstances, it can scream, "I'm a tool."
Now, if all your clients are 19 year old college kids, you're right, they probably don't care what you look like. But when you start dealing with 45 year old businesspeople who own $300,000 aircraft, they're going to wonder about shorts and t-shirts. Those are the clients who will provide a decent living for the CFIs who want it.