I think you are maybe also not thinking of the degrees of privacy people value.
For example, I used to have a job where it was valuable to be able to present to new professional acquaintances as politically neutral or at least politically agendaless. I have a very Google-able name. And Google really likes Facebook results. Therefore I kept anything publicly available, including on Facebook, fairly neutral—no tags in public contra pictures, for example. My bar was, if someone were to see they were going to meet with me, search my name, and read whatever struck their interest for a few minutes, they wouldn’t have significant information on my politics.
That’s a very different goal than keeping my information private from, say, the law.
I’m glad this worked for you, but would your thought be to use unique signs for each kid if each had a multi-month signing phase?
In particular, I would not use this approach too extensively if your kid may want to be able to communicate with others who work with kids—teachers at daycare, speech pathologists, many nannies, other pediatric medical professionals etc. I do agree that straight ASL isn’t quite right either. Our kid’s speech pathologist uses a lot of signs but chooses for example to use “car”—a fairly easy sign—for all vehicles since bus, train, etc. are more abstract or complex. This approach has allowed our kid to communicate with a range of people over the relevant time period, not just our household.