The short answer to your question is no, the many different language families aren't evidence that language was developed many times over.

Whether or not humans developed language more than once is a question that probably nobody has a real answer to. There are a lot of theories about the origin of language, but as far as I know, none are significantly more supported than any others. The most dominant theory is probably a lack thereof- a lot of linguists will say that tracing back the origin of language is impossible.

Tangentially related, if you're interested, is a talk about the origins of Indo-European, which goes into the difficulties of tracing back the origins of a language even that far; Indo-European was probably spoken about 5000 years ago, and most estimates of when humans first started using language place it about 100,000-200,000 years ago.