Hospital steps is not a big royal tradition.
Princess Anne was photographed leaving with one of hers (with a nurse carrying the baby)
Diana Princess of Wales posed, as did her SiL Sarah Duchess of York, as did the Duchess of Cambridge.
And that's it
Refusing to tell the press where they were, slipping out and avoiding any speculative press pack (if there were one, camping out when you know where the new mother is one thing, but covering two or more multi-entry hospitals on the off-chance is quite another) would be easy - they can say that details will be kept private.
So it would have been very easy to announce the onset of labour (even though I think its weird to have agreed to that bit) at some point during the labour, or if that were not done in time, to dump that bit of the plan. And then announce the birth.
Personal announcements, such and birth and marriages are not made without the explicit consent of the individuals concerned. The press office does what those individuals instruct. Harry fucked up the instructions (and may perhaps have been sulking if he wasn't totally happy with the advice on birth announcements), timing fuck up followed, but he seeks to blame "the palace" rather than his own inability to issue proper instructions (something that would only have a taken a word to his private secretary, so even with a brand new born, it didn't take much effort to do it right - indeed as an announcement was made I expect doing it right took the same effort as doing it badly)