Spider Surprise For Parrot

I am trying to be more spider-tolerant. This is not a default reaction for me, but since they are harmless, as long as they aren't on me, I try to let them live. I had a challenging encounter yesterday! After seeing this one on my wall, she found her way here:

I was so careful, going into the kitchen to brush my teeth, and trying to give her enough time to find another place to go.

And then I walked in on Beeps, right after he'd taken a bath in his water dish.  Of course, I had to rinse out and refill his water dish, so I went into the bathroom and flushed out the spider!

She was really wet, but OK.  I managed to get her outside, alive.

I'm not sure that was the right thing to do -- can house spiders survive outside?  At least it is warm out now, and hopefully if she doesn't survive, she was food for someone else.