Why does he care if your bag has wheels? If you with your bag (wheels or not) can walk or run as fast and as long as he can with his bag, then why do wheels on your bag matter to him?

Wheel hardware takes up space and weight. Many people here prefer wheels anyway, but an alternative is a backpack type bag (less weight, more capacity due to wheel hardware not taking up space), which can be squished into "slightly too small" carryon spaces or sizers, and may be less likely to be forced to be gate checked.

Wear your running shoes (you may need them to make tight connections, especially if your incoming flight is late), and pack your dress shoes. Fill the dress shoes with socks so as not to waste the space inside of them.

Do a web search for how to pack a suit in a carryon.