Almost the opposite. When straw poll voter are made and reasons are articulated for those votes in conferences of the justices, the process starts with the most junior justice and works its way up the ranks of seniority, precisely in order to prevent more senior justices from swaying the opinions of more junior justices.
The primary reason that the Chief Justice is different in role than the other justices of the U.S. Supreme Court and that this post is approved separately by the President and Senate, rather than elected by the justices themselves, is because it carries with it administrative roles. The Chief Justice is the CEO of the federal government's judicial branch administratively, as well as being a judge, and is responsible for proposing the judicial branch's budget and annual report to Congress each year, for personnel and facilities management, and for supervising the never ending process of revising federal court rules. It is a lot of extra work for a very modest additional amount in his paycheck every month.
The Chief Justice is, in fact, often not the swing vote in the Court. For a long time, for example, Justice O'Connor, and later Justice Kennedy, held that role.
Nonetheless, it is commonplace to mark eras in Supreme Court history by the period that a Chief Justice is in office, a bit like reginal years in a monarchy. This is, in part, because many Chief Justices, as firsts among equals, are known for having set a tone for the court while they served. But, it is fair to say that under the new 6-3 conservative majority, that Chief Justice Roberts is no longer serving that role.
Many observers see Chief Justice Roberts as the justice who, in part, by virtue of his additional administrative duties, is more concerned about the Supreme Court as an institution, rather than merely being concerned about the outcome of particular cases relative to his judicial philosophy. But, a majority of the court does not seem to place a high priority on preserving the legitimacy and role of the institution relative to outcomes on the merits. Prior to the recent shift in personnel on the court, he was more able to appeal to other justices from an institutional perspective.
