The Burke and Wills expedition were the first white explorers to cross Australia from South to North (well almost, Burke and Wills were balked by mangrove swamps on the Northern shore and didn't actually reach the sea) but that wasn't the purpose of the expedition.
Robert O'Hara Burke, in addition to being a poor leader and a terrible bushman had a major penchant for self-aggrandizement. The purpose of the expedition as laid down by Royal Victoria Society which was employing the Victorian Exploring Expedition and funding it from a combination of public contributions and government funds was to explore and map the aboriginal territories north of Coopers Creek.
It was Burke who decided that he was in a race with another explorer, John McDowall Stuart, who was working out of Adelaide, and insisted on pushing through to the northern shore of Australia.
It should be noted that of the four members of the expedition who separated from the main party at Coopers Creek to push north, Burke, Wills and Gray died after returning to Coopers Creek but John King, the most willing to seek help from local aborigines survived.