Pirrie had a more than a cold. He'd been unwell for weeks and in February had an operation for prostate cancer. In those days, the operation was probably more dangerous than the disease and his recovery was slow.
Pirrie didn't normally go on maiden voyages. There were too many of them. There is some evidence that he wanted to go on Titanic but was advised not to. Andrews didn't normally go either, but on this occasion he did. It's not known when it was decided he should go. Pirrie was in England at the time.
You says Anndrews didn't really went on maiden voyages. Yet he did so also on the Olympic, in 1911, and then his team counted Wilding also. So Andrews surely didn't traveled on evry ship, but he took part as a leader in both Olympics and Titanic ´s first crossings.
I wonder if Carlisle might have took part in too, if not resigning in 1910.