My 1967 Peugeot UO-8 frame was of steel close to mild. That bike needed realignment after every spill, even very minor ones. I usually didn't and the wheels operated in planes not remotely related to each other. Still, that frame went 19,000 miles before the right chainstay broke. (A failure that was perfectly safe and left the bike still rideable.) So, in answer to the OP's question, yes, mild steel is perfectly safe as a bike frame material as long as you stay with tubes with appropriate wall thicknesses, ie weight.

Ben