Your reading of the situation is essentially correct. Generally Cincinnati is regarded as the leading center for Bronze Age Archaeology in North America,and has been pretty dominant in the field for most of the last century. If you were committed to pursuing Bronze Age Studies for the PhD you'd probably want to apply to Cincinnati, Penn AAMW, Berkeley AHMA/Classics, MIchigan and UNC - these are probably the leading department for Bronze Age stuff. Places like Brown or ISAW (NYU) would probably also appear on the list, but would allow less specialisation. It is hard for Bronze Age people to get jobs period, but Cincinnati and Michigan in particular have decent placement rates, although the pandemic will clearly screw things up even more.