Yet their gif of a stiff-legged deadlift has the lifter bending his back.
Most fitness trainers went through a crappy class/college program and know less about fitness than the average fitness bro, let alone anything about muscle imbalances. At least, in my experience from paying a couple.
Running shoes are fine, but after you stop rotating at the hip, you've reached the bottom of the stretch for the hamstrings, and everything else is just stretching the lower back for no apparent reason; could lead to a bulging disk.
Running shoes are a bad fit for it because they both alter the mechanics of the lift by raising your heels above the toes (which is why you shouldn't deadlift in oly weightlifting shoes with thick heels either) and use amortizing cushioning that compromises your stability. You are better off using something with a thin flat sole (think chuck taylors) or just taking off your shoes.
As an aside, exrx is my go to site for no frills "non-bro-science" advice. They back it with simple old-fashioned gifs for each exercise.