Could BP have plugged the oil gusher sooner?

This is the second attempt by BP to cap the oil leak at the bottom of the Gulf of Mexico. The first attempt to put a cap on it didn't work because ice crystals formed in the line that was going to collect the oil and take it up to a tanker on the surface. The ice crystals clogged the line, so no oil could get up to the tanker, so it was a failure.

Is it just me, or does it seem like BP is only interested in collecting oil from that well? I mean, if ice crystals formed and plugged up the drain, that would have stopped the oil, wouldn't it? If so, and if your aim was to stop the oil, then it would have worked! But they couldn't transport the oil up into their ship, so it was a failure.

If someone knows more about this than I do, please feel free to correct me here.