Yes, when elections are publicly financed, good candidates have a better chance of running viable campaigns against wealthy candidates who can self-fund. Generally, it takes a lot of money to run a successful campaign. Candidates who are not independently wealthy or do not command support from big-donor groups can get left out of the game. Public financing evens the playing field.
I am on the side of agreeing with the fact that elections should be publicly financed. Why not be able to accept financial help from whoever is willing to provide it and supports the candidate? An advantage would be that the candidate is freed up from fund raising duties and can focus on pertinent issues.
The First Amendment guarantees free speech to every citizen. This includes free speech for those that run for office. It also protects free speech for people that want to express their opinions by supporting a certain cause, or supporting certain politicians. Publicly financed campaigns restrict how much money can be spent on an election. Placing constraints on how much money be donated or spent during a campaign is a restriction on freedom of speech. In short, no, elections should not be publicly financed.
Those who don't have the expendable funds to support a candidate are left out of the current process. How does a person on a fixed income when often the combination of monthly expenses and healthcare exceed the income decide which medication to live with out or which days of the week to not use the lights in order to contribute to a political candidate? This leaves the political process very much controlled by those who have and not those who are need.