To my knowledge, there aren't any North American spiders that would cause a "seconds count" type threat to your life. I suspect the way this normally works today is:

  Black Widow: Have a painful bite on the house!
  User: "Ouch!"  *slap*

  [one hour later]

  User: "Oh my god, I am in unimaginable pain."
  * User is driven to the hospital.

Or

  Brown Recluse: Have a painful bite on the house!
  User: "Ouch!"  *slap*

  [several hours later]

  User: This is starting to swell rather
        disturbingly, and smarts quite a bit.
  * User drives to their doctor

With snakes it's probably another mater. If a snake I could not identify bit me, I would be calling 911 first, and taking pictures afterwards before the ambulance arrived. If it's not a snake that I recognize as safe, then I wouldn't be taking the risk that it wasn't a venomous pet that escaped (since I can recognize the venomous snakes native to my region).