I just listened to a the presentation of Judge Barrett as a professor at Notre Dame referred to in your article.
Hardly a conservative political. She seems like a law geek who loves the study of law. For her, there are political conservatives and liberals and there are legal conservatives and liberals.
She is a conservative judge that closely follows the text of the Constitution or the statues passed by state and federal legislatures. On the other hand a liberal judge is one that starts with the text but then interprets the law in current times.
In the flag desecration case your would think she would she would be all against it since she is thought of as a political conservative. Not true. She opined that freedom of speech trumped(sorry) the destruction of the flag and was allowed.
Everybody wants to know about Roe V. Wade. She was very clear about how she felt about it. She is concerned about process not whether it should remain law or not. In her opinion ideally state law should be followed. If state law becomes so restrictive on abortion then the Supreme Court may reluctantly have to help make abortions reasonably available.
Above all she said the Supreme Court Justices should follow the Constitution and state law as and when written. She says judges should not allow political passion to affect their decisions.