How is it a leap? The fact that she was in her car instead of the pool area is only relevant in that it caught the employees attention. She brought suspicion of trespassing upon herself when she refused to provide her name and room number so it could be verified that she was a guest. What was the employee supposed to do at that point? She couldn't prove the woman was a guest so she followed company policy and asked her to leave. I could probably visit any hotel pool and find a working key card laying on the ground under one of the chairs.
This isn't a case of someone being harrassed in a public place or in their own driveway. She was on private property, had violated a rule by leaving her children unattended, and was refusing to provide information requested by the owner's representative. What did she think was going to happen at that point? They were going to just say "ok, nevermind, enjoy the pool?"
I still think the employee could have handled it in different manner and defused the situation but she definitely overreacted to being asked and escalated the situation.