I’m very happy to hear that the particular essential oil you use on your particular dog has apparently worked well for her. That’s great!
Still, alas, it isn’t a conspiracy theory that some essential oils can be deadly for animals. Especially smaller animals (e.g., a large dog will probably be better off than a small dog; a dog in general will probably fare better than a cat).
This is not about paranoia; this is about science. If you’re curious, you can check this study here, which found that approximately 1 in 10 pets who were poisoned by essential oils were not even given those oils deliberately by their owners. https://www.ncbi.nlm.nih.gov/pubmed/24344857
It’s unfortunate that a stranger you overheard one time was anxious about the smell of lavender — that’s too bad — but irrespective of said stranger’s personal anxieties, it’s still important for people to be aware that not all oils are safe for pets. Not even to inhale. Which is why I felt compelled to comment on the original story. :)