There really is no such thing as "making a game on Xamarin". Xamarin is really just a library that allows you write UI layers and share core logic.
In a game, you typically dont use any native UI components - you will load sprites and draw in 3D. So you will get no value from incorporating Xamarin. You could technically use image views as sprites and make a simple 2D game, but it wouldnt take long for the performance to drop to unacceptable levels.
What you are looking for is a C# cross platform game engine. This would allow you to work in C# but deploy to whatever platforms are supported. So if you want to get more value from your post the better question is
What other C# cross-platform game engines exist besides Unity?
Here's a decent list
https://alternativeto.net/list/6695/list-of-c-and-net-game-engines-and-frameworks
Many try out MonoGame, but it is more low level. The reason people like Unity is because they have thought through the hard stuff already for many scenarios.
Knowing what kind of game you are trying to make and what platforms you intend to target can also help get this question more useful answers.
You might be in the wrong subreddit. If you don't have a C# background and aren't looking to do a ton of studying to create that simple functionality, you may look at a no-code option like AppSheet:
Or you could hire someone and they could tailor a solution to your needs and give you invaluable insight on design.
I am glad that it did help you, can I ask your opinion on Web Atoms, JavaScript framework for XamarinForms ? I am author of this Product and would really help your views. You can try at https://www.webatoms.in/play
Evaluate Web Atoms, a JavaScript engine for Xamarin Forms, disclaimer, I am author of Web Atoms. You can put all your view models, Xaml and REST Services into TypeScript and TSX, (JavaScript and JSX) and your deployment/compiling time reduces to 99%, it is essentially React Native but with Xamarin Forms. You can explore demos at https://www.webatoms.in/play You can use code push (hot reload) on production.
I'm thinking about using https://platform.uno/ or https://www.nativescript.org/nativescript-is-how-you-build-native-mobile-apps-with-angular as the best two ideas now. Write once and have mobile app + mobile web available.
Thanks for your feedback, I'm glad to hear our experience is somewhat similar with yours.
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As for the tooling, we have been using Rider for quite a while now, and the majority of the team is really enjoying the experience compared to VS for Mac.
You can install Android Studio for the SDK manager GUI, you can use that to install the platform tools which contains adb.exe. If you don't want to install all of Android Studio, if you scroll to the bottom you can just install the command line tools, and use the SDK manager.bat to install the platform tools. https://developer.android.com/studio/index.html