Corporate interests, for one, generally don't produce great healthcare. The profit motive often leads to rushing the production with a lack of regard for safety or usefulness, leading to ineffective and even harmful products going on the market in order to maximize profit. Additionally, turning the race for a vaccine into a competition between companies slows the development that could be created if researchers instead shared research and worked together. After all, COVID-19 and other viruses are enough of a formidable competitor in these races without the quest for monetary capital getting involved.
However, governments aren't great at this either. Politicians tend to overly politicize health, often receiving money from corporate interests to pursue certain solutions. Additonally, they tend to do what they think their constituency, or whatever group they perceive to be 'the majority', wants, rather than taking into account a number of perspectives and what the people they are supposed to be serving actually want and need.
Ideally, something as urgent as a vaccine for a pandemic should be treated as an independent research project motivated by a desire for a good result, including perspectives, research, and results from scientists around the world. Experts in other areas, such as logistics and sociology, could also weigh in on how to most effectively distribute the vaccine. As for funding, given the interest in donating to COVID-19 research, the number of affluent citizens who consider themselves philanthropists and the near-universal desire for the final vaccine, I believe that it would be easy to get the needed funds for the sheer PR-stunt value alone. After all, despite the fact that most billionaires don't genuinely care for the public, a vaccine that could halt or slow a pandemic is certainly more memorable (and does more for one's public image) than something like Elon Musk's at-home flamethrower.