Depends.
Patients who undergo hemispherectomy at a very young age can recover remarkably well; a testament to the incredible plasticity of the developing brain.
https://en.wikipedia.org/wiki/Hemispherectomy
In adult brains, yeah, it’s a very different story. See also: stroke.
Honestly, you’re both right. It is only one of a multitude of highly interconnected interdependent organs; but it can (and usually does) do some truly incredible stuff while running within that system.†
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† Assuming, of course, that Descartes/Wachowskis aren’t right after all.<g>