Is it typical for the crew to have such a focus on the condition and status of a patient?
Life saving seems to be in the forefront of the decision making.
Yes to the first statement, you wouldn't take additional risks unless it was critical lifesaving.
Is hovertaxying in cloud inherently safe? No, of course not but the only additional risk of doing this compared to clear conditions is the limited visibility, there is no increased risk of aircraft malfunction just because it its foggy.
Being a SAR captain means you have to make decisions to balance risk against reward and every SAR pilot I know would accept extra risk if the casualty will die otherwise - it's not nice (hence the winchies concerns in the voice over) but it is a fact of life with SAR.
If you don't take any risks (with mitigation wherever possible) you won't rescue anyone because so many people get into trouble in difficult terrain and weather.
If you thought that rescue was tricky - try the same conditions at night..........