Yes, the Supreme Court has clearly become a tool of the right. It cherry-picks history for ideological reasons, disregards precedent, and turns GOP talking points into constitutional law. That record alone, however, is not the reason the public is starting to doubt the Court’s legitimacy. The more compelling reason is the absence of one or more swing justices.
As I noted in a previous post, the Court had a swing justice throughout my entire adult life until Kennedy retired. The presence of a persuadable justice gives hope to both sides and thus provides a source of public support for the Court. That no longer exists. Everyone knows where each of the justices stands on the major political and culture war issues; the outcome of any given case will be dictated by the identity of the parties. The only question now is how far and how fast the Court will feel comfortable in pursuing the reactionary agenda. So far, the answer has been quite far and fast, indeed.