Why does Swift use exaggeration in "A Modest Proposal?"

Swift's essay, "A Modest Proposal," is a work of satire. The dictionary defines satire as:

the use of humour, irony, exaggeration, or ridicule to expose and criticise people's stupidity or vices, particularly in the context of contemporary politics and other topical issues.

Swift notes that the state of the Irish population is "melancholy." Poverty is endemic and there is no work to be had. The people are starving, and many are reduced to begging just to feed their families for another day. These facts were the topic of much discussion at the time, and many potential solutions were bandied about in the press and in Parliament. None of them had been put into action with any measure of success, and Swift states that

whoever could find out a fair, cheap, and easy method of making these children sound, useful members of the commonwealth, would deserve so well of the public as to have his statue set up for a preserver of the nation.

He then drops the bomb of his "solution": the Irish should sell their children to butchers, and eat the meat. This will put money in parents' pockets, ameliorating their chronic poverty, and put food on the tables of thousands of households, thus preventing starvation.
He says that his solution really ought to be tried, since no other "expedients" are even remotely realistic, e.g. taxing Irish expatriates to raise money, only buying Irish goods to stimulate the economy, stopping the infighting amongst the various factions of the Irish population, or "teaching landlords to have at least one degree of mercy towards their tenants." By contrast to these pie-in-the-sky notions, Swift argues, his own proposal is eminently sensible.
Swift's language is deliberately light and breezy to reflect the uncaring attitude the English public has towards the people of Ireland. His blasé style makes his "modest proposal" far more shocking than a more serious screed could have, because the tone of the essay does not prepare the reader for its message.