The high level point is that many algorithms for which quantum speedups are possible can be reduced to the Hidden Subgroup Problem, which requires a few weeks of a group theory course to understand.
The Wikipedia phrases it backwards (as do you): "quantum computers" don't solve the hidden subgroup problem, the quantum fourier transform, which "measures" f in parallel, can be used to solve the hidden subgroup problem efficiently. The QFT is the fundamental thing, not the HSP, and it's the building block for basically any/all useful quantum algorithms.
Yes because we're talking about pure math here not popcorn and soda.
"Efficiency of hw arithmetic is not the bottleneck, lack of a poly time algo to factor prime numbers is"
&, the closely related
"Ability to reduce everything to HSP is not the bottleneck, lack of a scaleable implementation of qFT is"
It's not hard to explain the problem, as most math undergraduates will have taken some group theory course. But even "subgroup" means nothing to most computer scientists (speaking from personal experience interacting with computer scientist academics).