Does it sound natural and do you have any alternatives?
She and the government are not chosen by universal voting. They are, therefore, never eliminated by voting, so why would they care about the people's voices?

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In addition to what has already been said, universal voting does not sound normal to my ears.  We normally say: 'We do not have a say in voting her in or out'. Or 'she is not chosen by democratic vote.'

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First sentence is fine.
"They are, therefore, never eliminated by voting" isn't natural. Here's a few suggestions on rewriting it:
1: "They aren't answerable to the people, so why would they care about the people's voices?"
2: "They can't be voted out, so why would they care about the people's voices?"

My preference would be #2.

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Thank you!

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I would prefer "Neither she nor the government are elected. They are, therefore not eliminated/removed by voting..."