You're right that Orgzly is a native app, but you're spot on in that organice just happens to render nicely in a mobile browser. You also don't have to run a 'whole separate server' to use it.

organice is a progressive web app and specifically built for that purpose. More on this in the docs: https://organice.200ok.ch/documentation.html#orgcc693e5

As for the installation of it, there's no server required. More on this in the docs: https://organice.200ok.ch/documentation.html#orgfa38071

Having said that, please continue to use whatever works best for you^^ I just wanted to clarify these points.