I can see his point but is he maybe being a bit naive. Luxury is relative and what is luxurious in one country is ordinary in another.
There is also the problem of corruption in third world countries. How much of the aid actually reaches the people who need it.
Think about the old saying " as cold as charity". What is the motivation of the people giving.
I would much rather see wealthy countries increase taxation, particularly on the super rich and increase their aid budgets. Certainly give the aid money to organisations like Red Cross where there is less chance of the money falling into the hands of a corrupt dictator, and stop selling arms.
Without the sales of arms from western countries the current conflict in Sudan would struggle.

There is a site that grades charities [charitynavigator.org]. Early on I became interested in how some charities (mostly backed by religious groups) can actually cause harm. I was involved with a group Children International, that was run by a religion and, of course, no talk about family planning and schools were always religious schools. Another active group (even today) is Heifer Int'l. On the front it seems a good idea. But all the pictures with cute animals and smiling children are a sort of bait and switch. I did a little research and found a major problem. Most animals are eventually slaughtered. Most families receiving animals have no clue about animal husbandry and the animals are often placed in areas whose climate is not conducive for the specific animal. A lot of animals die and the people are left even worse than they started. Groups that offer food to the needy also create a huge problem. Food cannot be distributed to rural areas so food centers are set up. In no time those distribution centers become settlements. When people are well fed, of course they bear offspring who also need food so the problem is made worse. More food = more births which, in turn means more food is needed.

What Singer either overlooked was oblivious to is the fact of how the money you donate to charities is managed and how much actually reaches those in need. I worked for a medical research charity and had dealings with other charities. Before you donate money to any charity go their headquarters and have a look at the staff car park , it was an eye opener for me. There is a considerable amount of money wasted in the administration of some charities. I am not saying do not donate money to charity just pick your charity carefully.