I haven't been in this exact situation, but I've worked with swales, ditches, french drains, etc. I
think it sounds like it might work.
I would lean more towards a series of mostly level catchment pipes, as opposed to one large pipe draining down the hillside.
I would consider finding a way to put a valve/overflow on the flex pipes before sending water down the hill. If possible I'd maybe connect to the top of the flex pipes, so that water only flows when full and keeps sediment out of smaller pipes, before plumbing to your reservoir.
Depending on your reservoir size and rainfall, you may have to deal with diverting overflow.
Minimizing the overflow by keeping the water moving as slowly as possible, will likely allow for cleaner water and also help keep the excess uphill where your vegetation will benefit from it.