Saying that a quantum algorithm is "simultaneously sampling all possibilities and choosing the best one" has always been, I've found, a strange way of putting it,

Indeed, misleading and annoyingly common and the kinda thing that's always encouraging my more cosmic hippy friends down blind alleys. I'm hoping to find a better way, it seemed to me that MWI might have done that.

Maybe it doesn't, I'm certainly not an expert, hard for me to tell without being able to read a good one :)

This is better, certainly:

A quantum algorithm such as Grover's algorithm simply works by changing the probability amplitudes ... in such a way that the probability of the desired answer is much higher than the probability of any other answer.

Not far off my assumptions in the original request which is always encouraging.