There is no Astro Go compatible app for Chromecast.. Just AppStore (iOS) or PlayStore (Android.

Chromecast has its own dedicated apps, but still requires another mobile device to act as it's remote control. However, that other device ONLY acts as it's remote, but the Chromecast device uses its app (Chromecast enabled apps)  to directly connect to the service and stream. For instance.. Netflix.... You launch the Netflix app on your phone, connected to the same Wi-Fi network, both the phone app and chromecast device detects this and launches its Netflix app and streams directly. Processing is done using the Chromecast. AOTG doesn't have such feature as it's not a Chromecast-enabled app.

You can Cast your Android screen from the Google Home app... Though experience will vary and won't be as smooth as the Chromecast apps.

I can't remember at the moment if the app is compatible with the MiBox. But it probably isn't. Best would be to buy a Minix box like the Neo U1 or Neo U9-H.. No tinkering required, the app will just work. I know cuz I use the Neo U1 and the app works..Cheaper? Mecool M8S Pro L.. Though, some tinkering is required.

Personally, I wouldn't condone using the phone connected via MHL HDMI to the TV, but if that's your preference.. Can, cheaper to just get the cable or an adapter..