Bearfoil, saying "I have control" at the point things started to go wrong would likely have caused the FO (as PF) to relinquish the primary flight controls - the precise opposite of what Capt. Burkill was after. The infinitessimal issue that got blown up out of any kind of proportion was that the initial evacuate transmission either did not work or was not heard - it's not clear which, nor is it ever likely to be.
The scandal of the Burkills' treatment stems from the fact that BA issued the crew a blanket order not to talk to the press, but (and I mean this as no criticism of the FO) the press managed to pressure a member of the FO's family to talk him into providing an exclusive, whereas Capt. Burkill stuck by his instructions, causing the press to dig up any information they could get their hands on - most of it not relevant, a little of it not particularly flattering.
BA's public relations department was essentially caught on the hop by this accident - they said they'd handle the press and then clammed up. The combination of Capt. Burkill's statement which, like Sullenberger's, praised the whole crew and didn't aggrandise his own efforts - along with sticking to orders about not talking to the press while the FO did - gave rise to the false narrative that Capt. Burkill "froze" - whether this was an act of spite by the press or simple chinese whispers we're also unlikely to ever know.