Why I can only weight paint one bit of this mesh, but not the whole object?

Link for the “.blend” file download:

Obuolys17T_riggingEYE.blend (4.9 MB)

I want to paint weight on a particular part of the object, but it behaves weirdly, like you could see in this gif here:

Link to the gif - https://i.imgur.com/apYL4zL.gifv.

The object has rigs (4 bones inside), the rigs were parented with the object with the auto-weight function selected. This is my first time rigging anything, but looks like even after researching these things, I seem to run into problems that I can’t think of what it is causing.

try unchecking this button image

I’m soo f blind, thanks.