"Recipients cannot help being influenced by their donors"
You are trying to compare this issue with campaign reform and similar corruption issues. But there are at least two big differences: ease of impeachment, and number of issues.
It is easier for politicians to break their promises to the people once they are elected, because they know that it is very hard to be impeached. Also, everybody knows--people and politicians--that there are many issues representing many different groups of people. Some issues will receive greater emphasis than others, it is unavoidable. They can break promises on some issues, and offend some people and not others. This makes these politician's vulnerable to bribery.
By contrast, it is impossible for the FSF to break its promise to its people since there is only a single issue shared by everybody. Also, even if they did break their promise, the members can reject them immediately ('impeach' them) and join, or form, another organization.
In other words, any deviation from its single issue will be obvious and they would be universally censured. It is like getting a drop of tomato sauce on a pure white t-shirt--you can't miss it. If you have a complex t-shirt, with lots of colors and patterns, it is easier to hide a tiny stain.
But you are right about one thing, the FSF is doing some things for its donors:
https://www.fsf.org/patrons/benefits
