I assume you ask, "why not purple?" because purple is associated with royalty? The answer is that, until the invention of the artificial dyes we use today, red fabric was very expensive.

And purple fabric was EXTREMELY expensive. The only reliable way to make good purple dye, which was called Tyrian purple (so named for the city of Tyr), was to grind up a species of sea snail. You needed an astronomical number of snails to dye even the edges of a garment, such as the kind of toga a Roman consul might wear.

Red carpet was lavish. A modern comparison would be a private jet. Purple carpet would be like a private jet made entirely of solid gold.

Red carpet today is just, like so many other things, tradition.