In my opinion, wealthy interest that fund campaigns are not the problem.
It is public apathy.
By law, it is disclosed who give what money to whoever.
FEC.gov, and other sites have easy lookups for the campaigns.
It is up to the citizens to "follow the money", to see if the rhetoric and voting records of the candidates match their constituents or their donors.
It can be both.
Of course, it may not be, either.
The key is an engaged citizenry, to hold politicians accountable.
But, no, I believe publicly funded elections would have similar consequences, unless the citizens pay attention, it would make no difference.
Influence peddling would still going, and may be more well hidden than it is now.
Now, it hides in plain sight, should one bother to look, due to FEC regulations and laws.