Freedom of religion requires the state to carefully parse its authority to avoid threatening the moral foundation of the citizens. Her argument is false. Not every, or even most, cases involve moral challenges. When they do it is the responsibility of the government to insure its laws do not subject a preponderance of citizens to infringement upon their religious beliefs.
Her argument is very simple. Religion cannot be allowed to interfere with the authority of the state. That is a direct assault on the Constitution. Impeachment and removal are perfectly appropriate and necessary in Sotomeyer’s case.
That's the implication of her argument, you're right. But she doesn't even claim to make an argument - she denies it, in fact, by affirming a "perceived personal burden." That, in turn, has to be itself implied as referring to religious belief protected by the 1st Amendment. And then underneath it all is her fundamental implication that the government is superior to the people in all cases, and that that position is in fact the very purpose of government itself.
So much for "We the People."
I've read a lot of twisted legal arguments. I can confidently say, however, that her "question" is THE most twisted legal "argument" I've ever read. And because it's from a USSC Justice, it's all the more shocking.
America is in a LOT of trouble.