I assume that you are talking about the United States. If so, I would argue that we should not move to public financing.
I understand that public financing would seem fairer. If candidates had only a certain amount to spend, the richer candidates would not have an advantage over those with less money.
However, I think that Americans do have the right to use their money to promote their political beliefs. Therefore, I do not think the government ought to be able to limit how much money someone can spend on airing their political views. If we allow people to spend as much as they like on this, there is no point in public financing because people would just get around these limits by making "issue" ads that are really for or against some candidate.
The idea of public financing of election is designed to ensure that ail the parties have equal power to influence the public opinion through by ensuring that they have equal money to spend on their election campaigns. But this approach fails to address the basic problem of manipulation of public opinions using tricks of modern advertising and propaganda restricting the ability of public to elect candidates based on their true capabilities and sincerity.
Some people may argue that it is better to replace the greater evil of unequal use by money power by smaller evil of equal and controlled use of money power. However, by doing so, we tend to undermine and delay action towards total elimination of such evil.