I think the joke/soundbite example demonstrates a point: The Jersey Rule would be ok constitutionally. Even with that ruling standing, there is plenty of room for legislators to fix "money in politics" practically, without contradicting the SC. Laurence Lessig has some of the best known proposals.
The problem, as I see it, is status quo bias within any congress/parliament. Actually removing money from politics crashes their industry, puts all their friends out of work, and challenges their power. Every unsuccessful candidate, campaign manager, staffer, and such makes their living this way. Every successful candidate was successful at this way of doing politics.
It's a similar problem to legislation (or even customs) that would enable 3rd parties to succeed. It's against the interests of the Republican and Democratic parties, so it won't happen.
Not addressing the actual question that's being discussed here, but you can always amend the law
I'll less interested in sound byte-like fixes like "wear sponsorships on their jerseys" and more with what you can realistically change without overturning Supreme Court precedent or ignoring the Constitution entirely.