Is it because the modernity is defined through the extent of division of labour? Also, how tightly has this concept integrated with our current society and how is it affecting it? What measures can be taken to control it?
Modernity is a complex phenomenon coming out of the Enlightment and earlier roots; one aspect of it is the Industrial Revolution and the increasing pace of technology as identified by Heidegger.
The division of labour in contemporary classical economics is an outcome of the self-regulation of the market-place (and thus society) by the invisible hand.
This metaphor was developed by Adam Smith to explain the self-regulation of market-place; in all his works he uses the metaphor three times; first in his Theory of Moral Sentiments:
The rich … consume little more than the poor, and in spite of their natural selfishness and rapacity, though they mean only their own conveniency, though the sole end which they propose from the labours of all the thousands whom they employ, be the gratification of their own vain and insatiable desires, they divide with the poor the produce of all their improvements.
They are led by an invisible hand [emphasis added] ... and thus without intending it, without knowing it, advance the interest of the society, and afford means to the multiplication of the species.
That is the invisible hand is the best of all possible invisible hands; and it is thus through their natural and therefore rational selfishness.
The impact of this Smithian idea didn't become apparent until Arrow & Debreu turned it a mathematically model; where notions of commodity, price, rational selfishness and competitiveness were made precise. Like all models it had short-comings, which were easy enough to note, but it proved its strength by taking centre-stage in Economics. Its the thrust of these complex of ideas that form the core of the neo-classical synthesis.
Recent work has focused on a critical examination of its assumptions. For example, Stiglitz on imperfect information.
The division of labour is a description of how people can cooperate to produce more than they would without cooperating. To cooperate successfully, any given complex task should be broken up into parts that can be understood and implemented. For example, to make a computer monitor you have to make plastic and LEDs and ports for connecting the monitor to a computer and other stuff. You then have to assemble the monitor and ship it. If you pick one small part of the process of providing the monitor you can understand how to implement it really well and do a really good job of it. If you tried to understand the whole thing from start to finish you would fail because making the whole monitor and shipping it and advertising and so on is too complex. This has nothing to do with fantasies about perfect information. See Human Action, Chapter VIII Section 3.
You ask what measures are being taken to control the division of labour. Most governments make lots of rules that stop people from cooperating with one another, i.e. - from dividing labour in the way they see fit to divide it. One example: working for less than the wage that the constituency of some government official would be willing to work for is banned by minimum wage laws. the idea that the government can improve on the outcomes of the market presupposes that government officials somehow know better than the people involved in transactions. This is a curious assumption since those officials can't explain their objection to the satisfaction of the participants or they wouldn't have to threaten prosecution or fines to get their way.
You ask how tightly division of labour is integrated with our society. Understanding a task well enough to be able to get better at implementing is a requirement for the existence of civilisation. Without it no civilisation of any kind could exist. Every step to greater division of labour is an advance, as pointed out by Ludwig von Mises in Human Action, Chapter VIII, Section 5:
The division of labor splits the various processes of production into minute tasks, many of which can be performed by mechanical devices. It is this fact that made the use of machinery possible and brought about the amazing improvements in technical methods of production. Mechanization is the fruit of the division of labor, its most beneficial achievement, not its motive and fountain spring. Power-driven specialized machinery could be employed only in a  social environment under the division of labor. Every step forward on the road toward the use of more specialized, more refined, and more productive machines requires a further specialization of tasks.
The division of labor, as described by Smith, can lead to increasing returns to scale, which in turn means that it's impossible for each factor of production to paid its marginal product. This in turn implies that Euler's theorem alone can't account for the distribution of income. Insofar as the distribution of income is a central issue in economics, the division of labor is therefore of central importance.