So this is a workaround because it was driving me crazy too, but assuming you're on Windows I use nircmd.exe, a simple scripting tool, to move it out of the way. To do that:

Download nircmd.exe from here: https://www.nirsoft.net/nircmd/

Copy nircmd.exe to the same folder as your main Steam program (probably C:\Program Files (x86)\Steam)
Create a file called "RunSteam.bat" (or whatever you want to call it). You might need to make sure you have file extensions visible so that it doesn't turn into a "bat.txt" file. Right-click it and click edit.
Paste the following in the file:

"%~dp0nircmd.exe" movecursor 10 10
"%~dp0nircmd.exe" setcursor 5000 5000
"%~dp0steam.exe" -bigpicture

Save it and just run this file whenever you want to play Steam in big picture mode.
(The first command isn't necessary unless you're switching or disabling monitors to play in big picture mode, but it can't hurt. Basically something needs to "jiggle" the mouse for the second command to work, which basically moves it to the bottom right corner. If you happen to have a monitor with a higher resolution than 5000x5000, just increase those numbers ;) )

I use nircmd in conjunction with DisplayFusion to switch Steam to a TV, disable all the other monitors, get rid of the mouse cursor, change the sound from my PC speakers to the TV, then change it all back when I'm done playing and exit Steam. Pretty great tool!

Note that if you have Steam change to a preferred monitor when in big picture mode, the script above probably won't work. Switching to the monitor with DisplayFusion rather than the built in Steam setting gives you more control and will work (but unfortunately costs money), but the nircmd scripts need to run before Steam.exe is run. Theoretically you should be able to get around this by putting "start" in front of the nircmd commands, but when I tested it briefly it wasn't working for me.

Is this still the best way of doing this?
I'm new to Steam and Big Picture Mode and find it baffling the mouse cursor keeps showing in the middle of the screen when a mode made specifically for controller is activated.

I can see the utility in having the mouse be there for if you ever want to use the mouse instead but the immersion takes a pretty big hit as it is now icl.

My current settings are set to use Big Picture on a specific monitor (my TV) and I can confirm moving the mouse out of the way still brings it back next time "home" button is pressed or you back out to reach the "home" menu.

Surely this must be a bug?
Have they really just let it be like this?
Was the old Big Picture without this issue?

I'm grasping at straws here before I will have to resort to less official methods!