Hide glue intended for permanent construction (i.e., probably not something like fish or rabbit glue) is strong enough for many applications. It bridges gaps quite well, and rarely discolours the surrounding wood.
For a low strength application like mitred picture frames, it'll be more than strong enough. Not as strong as modern PVA glues, certainly, but for indoor low-strength stuff like this it'll be fine. Certainly, you don't need the gap-filling abilities of hide glue in this application: you should be aiming for nice thin mitred joints.
Most typical glues used in woodworking are "non-toxic" by any scientific measurement once cured, and that includes most of the PVA glues on the market. By the same measurements, few things in life are completely "natural", though I suppose traditional hide glues will biodegrade completely (but who knows about the convenience "hide" glues you can buy pre-bottled and ready to go these days). I assume PVA will degrade to tiny particles of "PV", adding to the cloud of micro-plastics we are currently swimming in.