So except for the recording sessions, you've done pretty much everything a curious client does to find out about their therapist (looking her up on Facebook, etc.). And everything she's done (lateness, slowness to respond, expects client to take initiative in contact during difficult times) is stuff people complain about with their therapists all the time. (Not to say it's not annoying.)

I don't know whether to have the conversation with her as you envision it, but if you go to another therapist, you should explain exactly why you left the first one to make sure the next one isn't habitually late, responds more quickly, and will reach out to you.