I don’t see how this works.

On one point:

The utilitarian argument construes the relevant ethical concerns, unsurprisingly, as utilitarian: the starting point doesn’t matter so long as the right results get over the line. This can be both one of utilitarianism’s greatest strengths and greatest weaknesses, and in this case the strength is that utilitarianism is highly accommodating of the fact that some but not all people are employees of Oxfam (or indeed any relevant charity or similar organisation). The obvious point to make is that If you’re not an employee of Oxfam then the utilitarian argument goes through, because giving to Oxfam is your means of getting those results over the line. If you are an employee of Oxfam, then perhaps you don’t need to give, because working for Oxfam is your means.

On another:

The sentence “his argument only allows for money going from non-Oxfam taxpayers to Oxfam employees” doesn’t include the important premise “the role of an Oxfam employee is to convert that money into good deeds done for the poor, for example by using it to pay for food in a famine”. The intended result is the same whether you are an employee of Oxfam or not (viz. paying for food in a famine). You want us to quibble about the wording (or rather: the wording as you have summarised it here) on grounds (which you leave implicit, so correct me if I’m wrong) that it is incoherent to say “everybody” when some people are already employees of Oxfam.

This seems to drastically confuse Singer’s actual aim (to convince the vast majority of people who are not Oxfam employees to give to Oxfam) for something not only very odd but plainly non-utilitarian, something like: “it is a deontological requirement that everybody give money to Oxfam”.