I've definitely had some of those issues. I won't count an old issue where my GPU needed a special connection to attach audio to its DVI output (rare oddity). Some others:

• Most computers would need to swap default audio device between whatever you use at a desk, and the TV registered as an HDMI audio device.
• Bluetooth connections to arbitrary controllers have gotten better, but they had often needed manual enablement each time through mouse-based menus or a number of firmware updates to work with Windows/SteamOS.
• My Steam Deck, even in its current iteration, takes some time to recognize the connected TV and swap resolution.
• The mouse cursor issue can come up if you had to do any mouse-based option swapping, like that thing with audio devices.

I've definitely gotten it working and had a blast, but the number of button presses to get to starting the game can sometimes be hard to predict. Even when I had a computer dedicated to the TV (a long time ago when SteamOS was fledgling) it was pretty unreliable about having all the right updates and not needing a mouse.