I am surprised no one has suggested your source is a weak link. I am a firm believer in the "garbage in is garbage out" in audio.
Yup. I agree @rikkipuu , once one has heard what a great streamer/player can do there is no going back.
A new or used streamer is where I would look to upgrade your system. If you can afford an Innuos or something similar I think it would punch the ticket for you.
He is in New Zealand, he has got to try Antipodes!
Seriously @OP even if one of their streamers are out of your budget at the moment, you must be curious how it might change the sound you get? I would love to hear your impressions- I may incorrectly assume it must not be hard to demo one if you are in the manufacturer's country? Are you close to a dealer who carries them, or are you able to travel there and demo one in their listening room?
And I also owned those Transparent Musicwave plus speaker cables a while back, yes, they were very average.