Of course, what you should do to make the world a better place depends on your ability. I'm of the opinion that you should help yourself first. Get yourself into a good place financially before you start donating, for example. Buy cheap stuff off Amazon and save up before buying from more ethical places.
The letter writer isn't in a position to hire POC into management positions and stuff like that, but say they walked into a store and saw a Black person being followed around by a paranoid white storeowner. That's an opportunity for the white person to speak up because they're in a safer position to do so.
Or, say the letter writer hears their uncle/grandpa/aunt/pastor/neighbour/other white person in their community saying racist things about Black folks. That's another opportunity for the white person to speak up.
If the letter writer belongs to a community like a church, they can also advocate for more charity work that benefits communities of colour.
Finally, poor people can vote too. They can vote for representatives in favour of anti-racist policy.
With regard to your second point, I totally see where you're coming from; truly, I do. There's a lot of guilt flying around and guilt is never productive. I think the problem is that "privilege" is still misunderstood as a concept. As the writer of this article explained, privilege in of itself is a neutral concept. It doesn't mean you've done something wrong or that you're a bad person. It's simply something you're born with and you should not feel guilty about it.
Privilege is also intersectional. Just because you have white privilege doesn't mean you have all the riches in the world. You can be white and poor, white and gay, white and a woman, etc. etc. That's why I made the point about myself being a queer person or colour WITHOUT white privilege—yet having loads of other privilege such as class/education/social/financial privilege.
You are right in the sense that part of the problem is the narrative and tone surrounding "privilege." The loudest voices that shout "privilege" aren't shouting at poor, marginalized men like the letter writer but powerful, rich men the likes of Harvey Weinstein who abuse their power. Unfortunately, poor white men like the letter-writer hear this and think it's directed at them. In a way it is because the letter does have white privilege, but his white privilege is very, VERY different from Harvey Weinstein style white privilege.
tl;dr—white privilege is a neutral concept and no one should feel guilty about it; rather, folks with privilege (any privilege) would do good work by thinking up creative ways they can channel that privilege.