I think part of it is the fact that it took a significant amount of time for the engineer to tumble to the fact that the ship would in fact go down . . . which explains why some of the adventurous passengers boarded rafts - and were later faulted for cowardice for not having stayed with the ship and gone down with her, giving place in the lifeboats to women/children.So that slashes the number of man-hours. What slashes it even more is that well over half of the people would have been women and children not able to do much physical work. And lastly, there was no plan and no tools in place to make use of the manpower that was available.
And as to making a refuge on the iceberg, the ship had no facility for deboarding onto the berg and no way to approach it safely without probably colliding and doing further damage, accelerating the sinking of the ship.
There are plenty of things that coulda, woulda, shoulda after the fact - but the bottom line is that if the ship you're boarding is gonna sink, don't board her. They should have had lifboats enough for the entire population on board, but they didn't. And even the ones they did have were, predictably, not all available due to the listing of the ship. Well, duh! - if you're gonna have to abandon ship it probably will be listing!
The fundamental problem was hubris. The ship "couldn't" sink, so why bother with lifeboats, and why worry about icebergs. It was all downhill from there.