The aging is a little strange. It is definitely based on playtime. The ages go; Baby -> Toddler -> Child -> Adult. If you do not have an empty play slot your son/daughter as an adult will just roam around your house and do nothing beneficial. It’s very odd in my opinion haha.
I think instead of RNG/luck, they should have set the price for each item in duckies so you can just purchase them when you collect enough. For example; common items cost around 75-100 duckies to unlock, and epic items (like the bathtub) would cost 200 duckies. Making events based entirely on RNG isn’t very fun :(