The short answer: No.
The longer answer:
To clarify this concern, virtually every bill that has been proposed since 2010 has a final section that reads:
"Nothing in this amendment will be construed as an infringement on the freedom of the press."
This is because people are free to use their own money to promote their support or opposition to a candidate, policy, ballot measure, or shoe color, all of which fall under First Amendment protections.
However, before corporate personhood led the Supreme Court to rule that money counts as the unlimited, even anonymous, free speech of corporations-as-people, the government (for, of, and by the biological People) was allowed to limit that spending. It did so in order to protect other freedoms, too, such as the freedom to have a government whose elected officials do not act in the interest of their private supporters when there is no equal way for the members of the public to compete with private interests. In other words, any significant imbalance in access to speech about and influence over candidates, elected officials, or ballot measures runs the risk of violating the freedom speech of those who cannot compete. As Justice Stevens wrote in the dissenting opinion of the Citizens United v. FEC ruling, when private moneyed interests are free to saturate our sources of information, "corporate domination of political speech during an election could impoverish rather than enrich the marketplace of ideas".
That is to say, when it comes to securing the public's good, a private interest's "freedom of speech" is limited, just as it is limited when we say one cannot frivolously "yell fire in a crowded theater". Freedom of speech is limited to provide for other freedoms. The public cannot enjoy its freedoms in the interest of the public good if ordinary voters cannot compete with the imbalance of accommodations for the richer and more powerful investors in elections.
Amendment language for publicly-financed elections seeks to protect the freedom of our electoral and political systems as public utilities or the common sector of the public good. An investment in a candidate or campaign violates rights, overstepping the bounds of free speech. Thus, ending corporate personhood is essential to winning back the right to a government that is , in fact, public, not private. Then, publicly-owned elections, or their equivalent in guaranteeing that no individual interest can outweigh any other in our system, can find their place in the Constitution.