It could happen, it just couldn't come out of the blue. Someone would probably need to pay for a nation-wide awareness campaign (the irony) about the evils of corrupt charities such as The Red Cross or Susan G Komen.
They actually only spend about 20% of their revenues on breast cancer research. They spend a lot of their money, about 40% of revenue, on "awareness companies" which have dubious real-world effectiveness, and can be considered a clever way for them to market themselves and fundraise without actually having to report it as such (they spend 10% on fundraising to begin with).
They're also very litigious. Of course they need to protect their brand, but in a lot of cases they are pretty objectively the bad guys. They have taken legal action against other orgs/charities using "for the cure" in their names, they've taken action against orgs/charities using "cure" and pink together.
In general I think they are more focused on self-perpetuation than curing cancer. There's a lot of reading material on how some of their corporate partnerships have been deceptive in the sense that Susan G Komen received very little directly financially, and took the opportunity as a chance to market themselves instead.
Their CEO and upper management do make a lot of money, but I don't think it's out of line for an organization their size. It's still a matter of controversy though.
They probably also haven't fully recovered from their brief attempt in 2011-2012 to stop working with Planned Parenthood [1][2], which is where a large number of women get medical care such as breast exams and cancer screenings. They had a huge drop in donations following that debacle.
For the curious, Karen Handel (now R-GA6) is believed to have been the driving force behind that.
They get a lot of criticism for (among other things) "pink washing" breast cancer, more of their funds are spent on "raising awareness" than actual scientific research, and having corporate sponsors that sell potentially cancerogenic products.
There's a documentary about it called Pink Ribbons, Inc.
I also heard someone once criticize them for supporting anti-choice politicians but I don't know how true that is.
Komen is very litigious when it comes to other non-profits using the color pink in their breast cancer awareness campaigns. They also pay their leadership a ridiculous amount of money but I'm not sure how it compares to other execs of non-profits
There are many arrows in the quiver that an enterprising politician can use to hold an errant organization to account (it's very difficult to be against holding powerful organizations to account, regardless of the actual merits of the action). They could propose amending the charter, simply choke off funding or insisting that they submit to a thorough audit by some federal government office (for which read: do what we tell you to, or have all your dirty laundry aired on cable news).