I’m not an OSMC guy, just an end-user, but I guess I can offer one user’s perspective…
There isn’t an app store, really. OSMC is designed to run just one app - Kodi. As Sam says, it contains a full implementation of Debian Linux, so anything you can run under Debian you can probably get working here, in time. But it isn’t really designed to run anything except Kodi.
You can enable some other functions by installing Kodi add-ons. For example, you can access YouTube or Netflix videos, but you don’t do it with a dedicated app, you use a Kodi add-on. That sometimes sets some limits on the experience - the UI is less sophisticated than a dedicated app, and there are often video resolution limits too. (Disney+, for example, is capped at 720p; Prime Video maxes out at 960x540). If you only want to play locally-stored video and audio, that won’t matter, but if you’re looking for a device to play Netflix, Prime Video, Disney+, etc. then OSMC probably shouldn’t be your first choice.
Streaming websites that you can watch in a browser. Is there any way to make them have a Roku like experience through some piece of software? For example, navigating Tubi or Crunchyroll like it is an App?
If your only looking at apps and streaming then buy whatever Android/Roku/AppleTV/Fire* that tickles your fancy. Or just upgrade to a smart TV that has the same type of device built in. You can do some of that stuff with Kodi but due to the smaller user base and DRM requirements there is going to be broader options on the other platforms.
Kodi’s strength lies in playback of local media files and provides a great interface for someone who maintains a library of their own media. If you own a PC you can just install Kodi on that and play with it to see what it is all about. If you find it is for you then you can then figure out what hardware and OS you want to run it on.
You can install a browser (but for that you need to install a Desktop/X-Server system) but Video playback in that browser would not be accelerated which would limit the Video Resolution you could play smoothly.