Because it’s a living language and still in use today. If I’m teaching something that requires an investment, I might as well be teaching something that has wider applications than just the first few months of life!
You could teach made-up or “homesigns” as they are called, but if you did, other people would never understand him. By teaching ASL, a non-signing caregiver can easily cross reference signs through various resources like ASL dictionaries either online or books.
The fact is that ASL has much wider applications than homesigns. Who knows, your baby might grow up and sign later in life with a deaf friend. Homesigns have no such application.
What if I want to sign with BSL or another country’s official sign language?
Go for it! There’s no reason you can substitute any set of signs in for this course, and for that matter, any other course. You can even use signs you make up on your own! However, keep in mind that your local sign language is best suited since you want your baby to be understood by the locals should they end up in daycare or want to pursue sign language into the future. Might as well start them off on the right foot!