Why can't we remember being babies?
From the BBC article, there's a prediction about that:

Perhaps, when we’re very young, the hippocampus simply isn’t developed enough to build a rich memory of an event. Baby rats, monkeys and humans all continue to add new neurons to the hippocampus for the first few years of life and we all are all unable to form lasting memories as infants – and it seems that the moment we stop creating new neurons, we‘re suddenly able to form long-term memories. “For young babies and infants the hippocampus is very undeveloped,” says Fagen.
But is the under-formed hippocampus losing our long-term memories, or are they never formed in the first place? Since childhood events can continue to affect our behaviour long after we’ve forgotten them, some psychologists think they must be lingering somewhere. “The memories are probably stored someplace that’s inaccessible now, but it’s very difficult to demonstrate that empirically,” says Fagen.
We should be very wary about what we do recall from that time, though – our childhood is probably full of false memories for events that never occurred.
Elizabeth Loftus, a psychologist at the University of California, Irvine, has devoted her career to the phenomenon. “People can pick up suggestions and begin to visualise them – they become like memories,” she says.