He was a complicated man. On the one hand, he liked to sleep on the floor without finery, and had to be told to stop because he's the Emperor. On the other hand, he basically waged a war of genocide against the German tribes. He clearly took personal development really seriously, but, like... genocide.
I've read him extensively and multiple translations. Your original post is strongly stoic, although the "tune out to it all and just live a good life" is a bit more Epicurean[0].
Stoicism is not about being a neutral perfectly calm emotionless robot. Stoicism is recognizing that you ultimately can't control the world, you can just control your (re)actions. The entire ethical system stems from that realization. Death or dying isn't "bad", it's just "not preferred", because the only good and bad things centre around things you can control. Everything else is just life. Roll with it.
(Personally, I'm with you though. I've learnt to eschew the labels and just lead a common sense life; but I do pick and choose from the valuable ethical systems over time.)
[0] https://plato.stanford.edu/entries/epicurus/