I don't think they forgot, but rather think they thought it would slow the whole thing down. Right now, the scene flows smoothly. Claire enters the room, sees the TV, sees her father down, and lunges for the telephone. If her surprise for Peter's survival had been left in, it would have had to be two scenes - Claire watches the press conference (presumably live, as the shooting would have been the big news and would have come first), sees Nathan talking, sees Peter standing next to him, and is shocked and delighted to see him alive. Then, Claire continuing to watch the press conference as the shots ring out, her shock that Nathan is hit, lunge for the phone. That would be a long shot of someone watching TV.

I wish they'd put it in somewhere, in her conversation with fPeter on the phone, or when fPeter visits her in California.
That's actually something I wish she'd caught. Not only is the phone answered by a man she thought was dead, but he also knows she's in California! That was a dead giveaway and I thought it would make her wonder how Peter could know she was in California.