International law clearly states that a sovereign nation has the right to self-rule, without external intervention. The UN Charter doesn't differentiate between democractic and non-democratic nations - it's up to the people of a nation to select their leadership.
We've seen this principle violated before, when the Ukrainian people took the streets for months to topple their leader in 2014. Russia to this day takes this as an excuse to question Ukrainian sovereignty, framing the events as a "US coup" to justify their violent invasion of Ukraine.
The argument you make just plays in their hand. "There was a violent coup - we need to remove the coup government and bring back democracy to Ukraine", they say. Because in your framing leaves open who gets to decide what it means to be democratically legitimized.
What if the US decides that it will not recognize the government of Denmark as democratically elected and moves to liberate the people of Greenland from their despotic dictatorship?
You're argument opens the door for unlimited military intervention.