Why is TI-Basic so slow?

It's slow because it's an interpreted language - Disadvantages of interpreted languages.

Basically this affects grabbing user input, as well as displaying graphics on-screen.

You could fix a lot of the code, like not using Goto, leaving parentheses off the ends of lines, not using ClrHome, etc. More info: https://en.wikibooks.org/wiki/TI-Basic_Z80_Programming/Tips,_Tricks_and_Optimizations