IMHO the role of stripped down OSes is one of the most overrated things - even in embedded system design. Basically it has next to no impact on performance and the only thing you gain is a smaller sized install image - which isn't really important if you install the system on a 8-32GB SD card. Sure, it's relevant if you only have 1GB of embedded flash or even less, but eg in case of RPi this isn't really a thing to worry much about.
If you compare eg Raspbian Lite to LE you'll notice the Raspbian Lite install image is about 360MB compressed and LE is about 150MB. But, if you look closer you'll also see that Raspbian comes with about 54MB (uncompressed) of kernel modules, LE with about 19MB (again, uncompressed) of them. Yes, they take up space but they might come in handy when you need one of those.
Then comes the big question: busybox or full-fledged gnu tools. Well, busybox is small but the tools also don't offer the full functionality. That caught the piCoreFolks, busybox modprobe didn't support the "softdep" option. Busybox ash isn't bash so even nowadays where most people are (or should be) aware of the term "bashism" you may come across some script that doesn't work with a standard posix shell because it uses commands specific to bash. Recently we ran into the issue where busybox's mount didn't cope too well with systemd. And so on and so forth, the list is basically endless.
So, when you design an embedded system / appliance the more important question you should ask yourself is whether you are prepared to put a lot of effort into maintaining a stripped down OS or if it's better to use some other system (like Debian), and eg just tweak the kernel and/or config settings a bit and make sure you install the packages you want and don't install those you don't need.
As you might have guessed there's no easy answer to that and both approaches have their pros and cons. But,, performance-wise both solutions will be about on par, most CPU time and RAM will be used by the main application - be that Kodi or some node.js scripts.
so long,
Hias