(Even though destroying a planet completely is utter overkill anyway. But I view the Death Stars more as mobile governing centres than just superweapons.)
Yeah, I mean presumably all life on a planet could be destroyed with just a few Star Destroyers and an extended orbital bombardment. But various rationalizations for the Death Star have been proposed over the years, including (1) some planets had planetary defense shields that could withstand an extended bombardment from conventional Star Destroyers (e.g. Hoth), but could not withstand the Death Star superlaser; (2) the whole point of the Death Star was massive overkill for the purpose of psychological warfare; or (3) the Death Star can also serve as a “mobile capitol” and administrative/military center as you said. I think a combination of all 3 rationalizations works pretty well to justify the existence of the Death Star story-wise.
Interestingly, regarding point 3, in A New Hope Palpatine doesn’t seem to be interested in relocating to the Death Star, and is instead content to entrust command to Tarkin. (But that’s really because at the time ANH was written, Palpatine was an ambiguous political puppet and an insignificant background detail, rather than a powerful evil sorcerer/mastermind.)