At first I thought you were joking, then realised you likely live in the US. I guess each country has it's share of things that other countries don't understand!
I don't know about whether they were joking or serious, but I think their point was that for any person, there are probably some charities they'd object to funding. I know people who won't donate to Habitat for Humanity (US charity that builds houses for poor people) because of their religious proselytizing. There are Catholic charities; there are charities that distribute birth control; there are even white supremacist charities. Aside from religious considerations, while charities in the US generally can't directly engage in political activity, there are lots of charities that are connected to political organizations, all across the political spectrum. Pick any controversy (climate change, gun control, religious freedom, reproductive rights, etc.) and I bet you could find charities aligned with either side of the question.
That's why they explicitly stated that they are excluding charities with religious and political affiliations. But there is only so much you can do to avoid controversies - there will always be someone who want to earn a quick buck on a scandal, or sometimes someone who just wants to watch you burn. Controvery is something people manufacture. At some point you have to stop caring.
There are issues that are inherently controversial.
For example (don't hate): if you believe human life begins and has value at conception, then abortion is in the same category as murder. If you believe it begins some time later, then any restriction on abortion is a violation of a woman's control over her own body and is in the same category as rape. Those positions are fundamentally at odds, and fundamentally do not lend themselves well to compromise. An organization with activities that touch on either side of that issue is going to be controversial, not because of religion or politics or making a quick buck, but because neither side is even remotely acceptable to the other side.
There are a handful of issues that are like that. There are a handful of issues where people are unwilling to compromise even a little bit. If you casually insert an organization that even remotely touches on that type of issue into a list of 30 organizations to support this month, with no ability to skip, you lose a certain category of sponsors.
The article mentions a 1994 incident in which a gunman murdered two people and injured five more in attacks at three different clinics that provided family planning services.
