As others have stated, streaming will definitely improve your audio quality.
I think that trying out inexpensive streaming options is a good way to get an understanding of what streaming is all about before dropping big bucks on a high end streamer. If you try out the Bluesound Node or even the $220 WIIM Pro Plus with the DAC in your Hegel 590, you will not only learn about streaming user interfaces, but you will also find out if your network is up to handling streaming, Chances are good that either of these options will sound better than using AirPlay. You can only go up in sound quality from there.
User interfaces for streamers are also part of the evaluation process for streamers. Some people, myself included, really like the Roon user interface. Since your Hegel H590 is Roon Ready, you can try out Roon for 14 days for free. If you have a computer that is connected to an Ethernet cable that can run Roon, you can install Roon on it and stream to your Hegel and see how it sounds. Using an existing computer to run Roon is not a long term solution, but the trial will demonstrate to you how Roon sounds and help you to determine if you like the Roon interface enough to buy a computer/server to go that route.
Good luck in in streaming journey.