Part of the relevance of Swift's "A Modest Proposal" is that it shows us that almost any ridiculous idea can be argued for with what seems to be "logic."
Swift's solution to the problem of poverty in Ireland is to sell Irish children as meat to be roasted, and their skins to sold to be made into leather. Of course this is absurd and Swift did not mean it seriously. Yet, his essay is a model of logical construction and argument.
One aspect of his "logic" is his use of statistics. Swift cites, among others, the following statistics:
a) the cost of raising a baby until the age of one year;
b) the total population of Ireland, and the percentage of whom are of child-bearing age;
c) the number of families who can support their children;
d) the average number of miscarriages;
e) the price that a young child could fetch if sold as a slave.
One could say that Swift's argument is "well-supported by statistics"; except, of course, that the argument is patently absurd.
Swift also uses the "logical" technique of discussing alternative solutions to the problem and showing how they are impractical.
He also makes a list of over six advantages that his proposal has.
Swift closes his essay with a bit of humble open-mindedness:
I am not so violently bent upon my own opinion as to reject any offer proposed by wise men, which shall be found equally innocent, cheap, easy, and effectual.
The next time you hear a politician or radio host recommend a new policy, think about Swift's "Modest Proposal" and ask yourself: Is the policy truly sensible? Or, has the person proposing it merely clothed it with some statistics and pseudo-logic?
Concerning your question about the relevance of Swift's "A Modest Proposal" in today's world, I'm not sure there isn't any place in the world where it is extremely relevant today. Are there countries that do not have people suffering from poverty? Are there countries in which landlords don't charge high rents? Are there countries with no corruption in government? Are there countries in which special interests do not successfully influence policies to be enacted that are detrimental to the poor?
The essay is certainly relevant in the U.S. today. Could anyone argue that government policies, as well as practices of the wealthy, do not contribute to the poverty of Native Americans and other minorities? Are there no exorbitant fees in hospitals? No overcharging by lawyers or big business? Is there no one in America that bilks the poor whenever they get the chance?
And the U.S. is the wealthiest country in the world? Is there no need for Swift-like corrections in Asia, the Middle East, or Africa?
Swift's outrageous proposal was, of course, meant to shock and to point out mainly how the British had exploited the Irish; in essence, he mocks the authority of the British officials. Later in his satire, Swift proposes reforms:
...Of introducing a vein of parsimony, prudence, and temperance. Of learning to love our country....of quitting our animosities and factions...Of being a little cautious not to sell our country and consciences for nothing...
Do these words not have a ring of our times? (e.g. partisanship, politics, Wall Street, banks)
It has been argued that Swift's main target was not the conditions in Ireland, but rather the "can-do spirit" of the times that led the people to devise several illogical schemes that would purportedly solve social and economic ills. It can be argued that the United States has done likewise: Today in the U.S. there are proposals that began with FDR that have led to the giant welfare state and debtor nation that this country now is.
Another point relevant to our times is that Swift's satire addresses the fact that England was denying Irish citizens their natural rights and dehumanizing them as a mere commodity. Here in the United States, there are certainly many who feel that their needs and wants are not being addressed and that they are being exploited in having to bear the burden of supporting so many who do not pay taxes, or are not even citizens.
The answer to this would depend a lot on where you live and what your politics are. However, in general, it seems to me that you could argue that this essay remains relevant in places where there is a great deal of poverty.
The specific conditions of Ireland are probably not really seen anywhere in the world today. However, there are many places where some people are rich and many people are poor. You can argue that the essay applies to those places. You can argue that it says that the people who control those places should pay more attention to the needs of the poor.