OP,
Please don’t feel that in order to get “audiophile” quality sound you need to spend $500+ on a Bluesound Node or Aurelic Aires Mini. These are great devices, but they are more than twice the budget you stated of 200. People on forums such as this have a nasty habit of ignoring budget and simply recommending one of the devices they have experience with and like, or worse, something they’ve read about and which is well reviewed. (In the case of the latter, all they are doing is serving as an echo chamber.)
There are three functions you need fulfilled by one or more products
1. Software that can stream your music from your devices (PC and mobile device) to your network, in other words server software
2. Hardware that can receive your streaming music from your network and turn it into PCM signal, either TOSLINK, SPIDF, or USB
3. Software that you can use to control what you play (you will need this on both your PC and your mobile device
You may already have 2 in the form of your chromecast. I’ve never used chromecast myself, so I can’t advise you what server and playback software are compatible, but I’m sure you can research this.
Airplay is a viable option, assuming you are running software on your PC that supports it, but as far as I know, the only software that does is iTunes. Also, the Airport Express, which performs function 2, only offers TOSLINK out. If that works for you, great. From a SQ perspective, Airplay can be improved upon. It relies on your playback devices (PC / mobile) to decode the music files to PCM, and streams the PCM. So a spike in network activity can cause your music to freeze. Also, jitter is quite high.
If chromecast and Airplay also won’t work for one reason or another (e.g. it isn’t compatible with your DAC, or you want to stream hi-res tracks and it only supports resolutions up to 48k/16), or you better sound quality, then I suggest you check out what can be done with DNLA / UPnP. Tons of devices support these protocols, some of which you might have lying around without realizing! e.g. almost every network DVD player made in the last 5 years can be used as a DNLA / UPnP streamer. And there is plenty of free or very cheap software that can perform functions 1 and 3 available for Mac, Windows, Android, and iOS. If you don't already have a DNLA / UPnP device on hand, then for well under 200 you can build an audiophile grade DNLA / UPnP streaming device using a Raspberry Pi, which I'll describe in a moment.
For function 1, options include JRiver, Kodi, Plex, anything from this page: https://en.wikipedia.org/wiki/List_of_UPnP_AV_media_servers_and_clients
For function 3, JRiver and Kodi both can perform this as well, and both have their own remote clients for mobile devices. I happen to like 8 Player for iOS. There are tons of other free options, so try out several and pick the one with the functionality and interface that you like.
For function 2, if you want to build a dedicated DNLA / UPnP streamer using a Pi, start by picking up a Raspberry Pi 3B, and get an 8 or 16 GB microSD card and install your audio playback OS of choice (free options include Moode and Volumnio, JRiver and Roon both make software that costs money). Also buy a power supply (you can go cheap switching supply or expensive linear power supply) and Hifiberry Digi+ or Digi+ Pro (to obtain PCM output over TOSLINK or SPIDF — otherwise the Raspberry Pi can already already give you USB output), and finally a case that is sized for the RPi together with the Hifiberry (HifiBerry makes some of there own, but there are others, or you can improvise. If you plan to connect the Raspberry Pi to a wireless network, be sure to use a plastic case, since a metal case could degrade your signal. If using wired ethernet, then you have no such problem. The solution I just described costs less than 100 and will give you sound quality that will rival and possibly exceed the Bluesound and Aurelic Aires Mini.
If you decide to go with the Bluesound or Aurelic, you are getting good products that will provide you with reliable streaming playback. Without a doubt, the biggest issue with the DNLA / UPnP improvised approach that I described is that it isn’t a commercially built and debugged all-in-one solution. There will be times when your server software crashes, or when your RPi freezes, or whatever, and your improvised solution needs a full reset.
Goodluck!