dah dit dah dit - dah dah dit dah
The reel does only hold the line,
most of the time, so one tends to take it for granted. You don't need to spend hundreds, just enough to ensure that it feels good for you and works when its required to - both of those 'qualifiers' can only be answered by you

Is it light enough, wide enough , smooth enough, have a drag system to suit the style of fishing and ergonomically operate in a way that suits your fishing style - Because you will take it for granted, it's pitfalls will only become apparent when it is needed, by which time it will be too late.
So the answer regarding reels is both yes, and no.
The rod..... Well the lady of whom you speak is Joan Wolff
'she said the line is probably more important than the rod or reel' - That piece of youtube footage is taken from a more comprehensive film that she made called 'The Dynamics of Fly Casting' taken by many to be a leading piece of media in the instruction of fly casting. It cannot be taken in isolation and Joan herself demonstrates casting with a Hazel stick. The action and weight of the rod are down to your personal choice, you may be suited to one rod and not another, a bit like shoes - you try them before you buy them and sometimes you find they give you blisters. So a £20 rod may do all of it that you ask and satisfy yopu for the rest of your life, or you may search and search until you spend £700 - or vice versa.... On the other hand you may meet the middle ground with a rod that suits your ability and style. To be honest the scale of rods is infinitely variable as is the style of each individual caster.
So again the answer to your question is both yes, and no.
73's