I'm sure that parents have been asking themselves this same question since babies were invented. I think you did fine, so don't be too harsh on yourself.

I think there are certainly situations where taking a moment to stand back and assess things would be best, but there are certainly other situations where reacting as quickly as possible would be the best. E.g. your child is in the street when he cries out and could be run over at any moment by a car.

I would say that there is no right or wrong answer because every situation is different. However, the ability to transition between them if you find yourself in the wrong "mode" for the given situation is a handy skill. E.g. "charging blindly" at the first screams but then quickly switching to an observe-assess mode in the snake example you gave once you became aware of the snake.

There are certain situations where you could certainly try always follow a more measured approach, but those situations are generally rather specific and specialized. E.g. first aid situations could benefit from certain decision trees that force you to slow down (e.g. ABC's, LAF-look, ask, feel, and so on) instead of just instinctually doing something that might hurt the child even more like instinctually picking them up and hugging them close.

But the range of possible trouble scenarios involving a child are so varied that I don't think anyone can or should advocate always doing a measured vs more emotional response. And it's certainly 100% natural for any parent to almost blindly go to their child's aid, and we humans haven't died out (yet) so it seems to work all right for us.