I am a "mature" teacher of mathematics in a conservative boys' school. I feel duty-bound to be a role model to my students in fashion in addition to the other responsibilities. For many years I have taken the stance that if a man is wearing a tie he should be wearing a long-sleeved shirt. I see many responsible citizens bucking that convention, and I note that many quality shirt manufacturers market short-sleeved shirts obviously designed to be worn with a tie. What are they thinking?
The existence of the tie in Western society dates back to the early 1600s. The Croatian military would wear small handkerchiefs knotted at their necks during the Thirty Years War - a look that impressed the Parisians who then took the look back to France as a fashion trend. Elaborately knotted cravats were popular until the Industrial Revolution, when men needed some sort of neckwear that was easy to put on, would knot firmly and would stay all day. The modern necktie as we know it was born.
Contrary to your observations that short-sleeved shirts and ties may be becoming acceptable, go with your gut instinct - they are not OK.
GQ men's magazine, Richard James of Savile Row and countless other stylish and knowledgeable people agree that short sleeves and tie does not make for a harmonious pairing. Charlie Porter of The Telegraph agrees it's not just tradition that makes it unacceptable: "There is something about the cut of a short-sleeved shirt that lessens the suit's impact. Without shirt cuffs visible underneath, at the wrist, a suit jacket loses a visual anchor point. Take the suit jacket off, and the sleeves of a short-sleeved shirt stand away from the body like wings, rather than sitting close to the arm."
A short-sleeved shirt and tie will never make you look elegant, intelligent, powerful or impressionable. It will make you look like a character from Napoleon Dynamite, a cast member from a low-ranking American sit-com, or - rather worryingly - a stereotypical mature mathematics teacher.
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