"Anything else faster/easier/gentler than needle nose pliers?"
My favorite on delicate fish is a kind of forceps, aka "roach clip," durable stainless steel, no-slip jaws, clamps shut. I keep one on a lanyard on my fly-fishing vest. Sometimes even barbless hooks are hard to remove and as long as you can see the hook, the forceps make its removal simple.
The plastic tool you show works even if the hook is swallowed so far you can't see it, but it needs to fit the size hooks you're using. My experience is some hooks are too small for even the small side. No, it's not especially gentle.
A link to one source of the forceps:
https://www.amazon.com/Eagle-Claw-Fo...h+hook+remover