(Although I hope Michael Bloomberg throws his hat into the ring. Whatever else you might have to say about him, at least he wouldn't have to take any money from anyone to finance his campaign.)
You could say the same thing about Trump, really.
It doesn't avoid the problem of why we don't want a plutocracy financing political campaigns. They aren't taking orders from big money; they are big money.
Except that Bloomberg a) is way richer than Trump, b) didn't inherit his wealth, c) has never filed for bankruptcy, and d) is closer to the centre than Trump. And I'm not necessarily thinking about his going for the Republican nomination--he might just decide to run as an independent. A sane Ross Perot might have a decent shot (or at least do to the Republican candidate what Perot did and lock up the election for the Democratic candidate).
Technically, Trump never filed for bankruptcy. One of his sub-corporations did. As much as I hate the guy, there is a difference (bankrupting a sub-corporation can be tactically smart as the law allows the share holders and owners of the sub-corporation limited liability).
And yeah, Bloomberg... The same guy responsible for ordering New York's police force to brutalize occupy protesters.