But is Oxfam really concerned about Americans earning slightly less than average Americans rather than people in desperate poverty in other countries?
No they aren't. If you look at their "finances and accountability" pages, you can see where they are spending their money. https://www.oxfam.org/en/what-we-do/about/our-finances-and-accountability The "Pride Guide" which had the FAQs is unreferenced.
Top 5 countries where money is spent are: South Sudan, Iraq, Democratic Republic of Congo, Yemen, Syria. Now I'm sure that being a lesbian woman in Yemen or a gay man in Syria is not a bundle of laughs. And yes, gay, lesbian and transgender people are discriminated against in Syria/Yemen/Iraq and that is wrong, but as well as being discriminated against they are starving to death and have no clean water.
Pride month is a very developed world sort of thing. Europe, North America, Australia, Brazil. Quite how an Oxfam in Chipping Norton displaying a pride flag is making any difference to a gay/trans person in Iraq beats me.
The problem is that most volunteers and members of the public perceive Oxfam as a disaster relief charity, which raises money to improve the lives of people living in abject poverty in places like Syria. To be there when there's a natural disaster to provide clean water and food. Oxfam Head Office want to be a political campaigning organisation.