I did a chair with a bent lamination back. It has 3/8" slats. They're glued with pva (titebond extend), and they're not cross grain like plywood. They have a reasonable amount of flex and you can feel it leaning back into the chair. That was my goal to let them move and bend for comfort.

I think a bed with 3/8" plywood would be stiffer than solid wood, but my biggest concern would be if you want it to feel like it's flexing, regardless of if it holds (I see it holding just fine). My thought process when I did the bed was I want a solid feel behind the pillow. If I'm leaning against the headboard I've moved the pillow to be a backrest and I didn't want the back to start flexing even more