I was a professional game designer (still a designer, but more a hobbyist now) with extensive (>10 years) game, mobile, full stack web and enterprise development experience, and all of that is current, as I'm developing games, including mobile games, in my spare time. It's very possible to do both.

I was one. The role doesn't preclude programming or software engineering experience--it very often includes it.

Satoru Iwata was a notable game designer. He was very much a skilled engineer, but his resume would've been trashed.