Is there currently or has there ever been historically a chief Rabbi of the United States? If not, why not?
There has never been an official Chief Rabbi of the United States. Jonathon D. Sarna (in his American Judaism: A History. Yale University Press, 2004, page 105) explains this phenomenon thus:
But since there was no parallel Christian religious authority—no chief Protestant minister, no archbishop, not even a Catholic cardinal with nationwide jurisdiction—it was easy for opponents to dismiss any Jewish effort to create a chief rabbinate as "ridiculous" and antithetical to American ideals. Nor, given church-state separation, could any chief rabbi expect governmental recognition, much less the kind of authority that some European chief rabbis wielded. Finally, many Americans of the day, Jews among them, were deeply suspicious of strong central authority. The "tyranny of the majority," they feared, would soon come trampling down on the rights of the minority, be they southerners or religious reformers. As a result, all efforts aimed at electing a chief rabbi ended in failure, and the decentralized congregationalist polity that characterized American Judaism since the Revolution remained in place.
There were a number of US cities that had Chief Rabbis, but the last one, Rabbi Shalom Rivkin of St Louis, MO, passed away in October of 2011 (source).
The point is that most Jews have not shown much interest in surrendering even a wit of their religious autonomy to their clergy. The only place in the world where chief rabbis have real political power and wield enormous internal communal clout is, of course, in the one place where Jews are in the majority – the state of Israel.
In the decades that followed the combative downtown rabbinical troika, supporters of esteemed rabbis lavished the largely honorific title of “chief rabbi” of specific hometowns upon their leaders. In the interwar period, Rabbi Tobias Geffen of Atlanta was revered both in that city and all over the South. In more recent times, Rabbi Pinchas Teitz, who unquestionably held sway among his minions in Elizabeth, N.J., comes to mind. But if they – and other compatriots – had any power, it was solely in the realm of persuasion.