Smartphones have changed this. You can write characters directly on a touchscreen. It's not as popular of an input method as pinyin, but it's not uncommon either.

There's also Bopomofo, as an alternative to pinyin input.[0]

0. https://en.wikipedia.org/wiki/Bopomofo

Writing characters on a touchscreen seems horrible.

You can write the most sloppy cursive Chinese character and your phone would recognize them just fine. Together with text prediction it's quite easy to get 40 characters per minute (probably equivalent of 80wpm in English), much faster than most people writing on paper.

Chinese doesn't have to use A-to-Z with Pinyin. There are input methods like Cangjie, Wubi, Bopomofo, 1-to-9 stroking on number pads, and handwriting. But yes, pinyin A-to-Z is the most pragmatic for many reasons.