I think the following is a good answer, and quite detailed. You can find the original here.
「死亡」については,「死」を客観的に捉えた表現なので, 仰る通り,「事故・犯罪」で用いる事が多いですね。
それと,ペットなどの動物に使ったりしますね。
「死去」は「死んで現世を去る」という意味ですが, 基本的に人にしか用いません。 一般的には,家族や親類縁者といった身内の人間や,
マスコミ関係が著名人の死を伝えるような, 不特定多数の人を伝達対象にする場合に用います。
So basically, in the word「死亡」, the character 「死」 objectively indicates the status of death and hence it is used in accidents, crimes, etc, as well as for pets.
On the other hand, 「死去」 bears the meaning of "dying and parting from this world" hence is basically used only for people. Normally it is used for family members, relatives, and in mass communication to transmit to a large audience the death of some famous people (so this is probably the case you mentioned about Chester Bennington).
Finally, 「逝去」as you say bears a sense of respect, and hence it is used for friends, acquaintances, or anyone we respected despite his/her being "great" or not (偉い・偉くないに関わらず). Anyway, it seems that you cannot go wrong in using such word even for people outside of your inner circle after all. The full quote below:
「逝去」は亡くなった方に対する敬いの表現なので, 偉い・偉くないに関わらず,友人・知人等といった,
身内以外で関わりのあった方に対して用いる言葉と 捉えていれば間違いがないかと思います。
たまにマスコミの報道でも「逝去」を用いる場合がありますが, その場合は,社会的貢献度が非常に高かった方のケースですね。
So as a last note you can also see that 「逝去」 can be sometimes heard in mass communication media, but in such case is most likely regarding someone who gave great contribution to the society.
To add more references, you can look as well here and here.