There is no such thing as "public money." There is only money that the government has taken from people via taxes. So, if the government takes money from me, under threat of force, and then spends it on a candidate with whose views I disagree, it has forced me to support a candidate that I do not want elected.
To agree and rephrase, "public funding" looks like incumbents controlling how campaigns are funded and how those funds are spent. Add that on top of the existing gerrymandering issues and it sounds like we'll have a government even less accountable to the average voter.
That sounds extremely un-democratic to me.