You are combining two things. "They were three"(which can only mean people) and "There were three of them." (which could be three of anything) You are making "They were three of them." It has to be one or the other. But you could also say in French "Il y en avait trois." (There were three of them)
I'm glad I only had to repeat this sentence. Colloquially, there is nothing wrong with saying, "They were three." I've heard the present tense version of that, "We are three", numerous times at restaurants in response to the query "How many of you are there tonight?"
As far as being about people, I respectfully disagree. Even the response "There were three of them," does not imply people. For example, "How many birds did you see?" "There were three of them." "How many bikes were there?" "There were three of them."
"They were three," however, does highly suggest people which seems to make it more correct if "ils étaient trois" actually implies people.