A couple of points.
This may seem a bit pedantic, but democracy per-se is not always a good thing. We use that word as a shorthand for "free societies that do things in a way that we think makes sense". Typically, that respect individual liberty, the dignity and value of the citizen, a certain amount of capitalism (face it, capitalism produces prosperity like no other system), and a certain liberality.
But democracy does not always lead to these things, in and of itself. When Palestine got to hold their first real vote, they chose to be ruled by a terrorist organization. Afghanistan is run by the Taliban because the Taliban has a lot of support among the population... or at least no competitor has enough support to challenge them.
Many Middle Eastern nations, if they allow voting at all don't allow women to vote. Many of these countries have demonstrated popular support for Sharia law, such as beheading adulterers, apostates and gays.
There's the old saying, pure Democracy is two wolves and a sheep voting on what to have for dinner.
The Founders, despite their flaws, were some pretty smart fellows. They did not want pure democracy.
We have a mixed system in the US, with checks and balances and different branches and so on. MOST importantly we have LIMITED GOVERNMENT... at least in theory. The Constitution remains important despite many incursions on its written limits.
Many nations have democracy but no hard limits on how much control government can exert, and their continued status as free nations is uncertain.
Another question is scale. In the US, a lot of things are settled at the state or local level. If one State congresscritter represents 100,000 voters, I might get a chance to talk to him on the phone or 15 minutes in his office. My chances of getting personal facetime with a US Senator, who represents millions if not 10s of millions, of people, is much more improbable.
When I vote in national elections, I am one vote among a hundred million.
Now imagine that scale of world government.... how far removed the legislators are from the common folk, how little the Senator from Bangladesh might know about life in Oklahoma before voting on an issue crucial to Oklahomans. How unlikely the population of Oklahoma is to make an impression on the World Senate (or whatever) and how it legislates. How little my one vote would seem to matter when billions are voting.
And remember the sheep and the wolves. There are MANY countries in the UN who would love to have the authority to plunder the wealthier countries for the benefit of their own.
These are just a few of the reasons I'm not a supporter of the idea.