Add function to all subsites who use different themes than main site?

So normally you would created a functions.php in your child theme to add your functions and if all your subsites use the same theme that works fine.
What if your users with sub sites switch to a different theme or install their own theme?

Then the functions.php would need to be added to each available theme?

So I could just create a plugin that was active on all sites instead of using the functions.php?

Then if I added a function later I would just update my plugin on my main site or site I used as a template in "new blog templates", and would that update my plugin across all my sub sites?