Would you enforce that on “friends of the campaign” as well? E.g. Political Action Committees.
John Jackson is running for political office using only public money. However, Big Titanium Inc. creates the privately run and funded “Friends of John Jackson” organization that is entirely unaffiliated with John Jackson but nevertheless runs pro-John Jackson ads in TV.
And if that's clamped down on, they just run increasingly vague ads that don't mention a candidate by name, but are clearly boosting the person/party/side they want to see succeed. A lot of people--particularly Democrats--like to say the Supreme Court got Citizens United wrong, but there's no way to get money out of politics while still maintaining any level of free speech.