Political parties are very interesting in the sense that they are not public; not really at all. (but they should be). That's an argument for stricter laws when specifically dealing with speech and campaigns.
Keep in mind, free speech has limits. Obscenity, classified material, threats of harm ... Free speech in the US, although more complete than many other nations, is still restricted.
Please no. At its core, a political party is a group of people with common goals banding together to contest elections. Congress and the elections are the public institutions. How people choose to form alliances and contest those elections is a private matter. The only way to make political parties public is to ban private political parties, which should throw up all sorts of red flags in your head.
As for Citizens United, people forget that it's not just about big for-profit companies spending money. Any non-profit group is a corporation. If you and I share a set of political ideas and want to advocate for them, we'd form a non-profit, print pamphlets, run ads, court donors, rinse and repeat. That's freedom at work. It's totally impractical to run such an operation as a single human being. It's totally irresponsible to set up such a group and then legally run it as a single human being. Incorporation is the legal means for people to embark on common projects.
[1] https://en.wikipedia.org/wiki/New_York_Times_Co._v._Sullivan
They just can't vote.
For example, while they can break the laws like people, corporations are not punished the same. When was the last time a corporation was "executed" (e.g. corporate charter revoked)?
But, in all fairness, that's not their primary purpose, it's merely a side effect of their overall utility. Most people in most corporations are decidedly not criminal, and they don't behave as criminals.
Once you give the government a tool to restrict speech / spending, it will be used to cut both ways.
Although donations to the wrong political group can certainly get you in big trouble today (if you're willing to stretch the definition of 'political group'), so...hm.
Hint: there's a reason the ACLU believes Citizens United was the right decision.