You've asked two different questions. You asked Why hasn't India been a more important global player throughout world history? The Answer to that is that it had been an important player through out world history an I can happily give you a run down of South Asian history in a global context. But you've also asked why don't we see Indian media or news globally and that's a different question to answer.

I'm not a historian, but as an Indian with an interest in my own history, I think I'll be able to provide a fairly detailed (if slightly biased) argument on why India was influential for a long time.

Firstly, the region known as the Indian Subcontinent has, for most of history, been fragmented into smaller states. The few times that it has been unified under a single banner, these were indeed very influential states. The first time that you can talk of a unified Indian state was the Mauryan Empire. The Mauryan Empire covered all of the subcontinent, along with afghanistan, eastern iran and southern uzbekistan. It was the reason for the popularization of buddhism in the subcontinent, which further led to the spread of buddhism into central, east and south east asia. The Mughals were another empire that conquered most of the subcontinent, and they, at their zenith, were the richest state in the world after the Qing.

Secondly, Indian culture and languages played a huge role in southeast asia. Even today this can be seen by the pervasiveness of sanskrit and tamil loanwords in filipino and malay languages. Most of the non latin scripts of SE asia are based on the ancient brahmi script in India. This influence in SE asia came mostly from trade and cultural diffusion, though some argue that there was a chola (old tamil state) conquest of the srivajaya kingdom in indonesia and malaysia. As to why these influences never spread westwards, I can only speculate and say it might have something to do the existence of the powerful unified persian state in between India and the middle east.

Thirdly, Prior to the industrial revolution, the region known as the India had the highest cumulative GDP in the world. This is mostly because of its massive population and fertile land, but also perhaps due to its role in the Indian Ocean trade. It was only after the industrial revolution that Western Europe came to have the economic dominance that it has today. And as to why the industrial revolution happened in Europe not in India or anywhere else, thats a question for somebody else.

In Conclusion I'm saying that the Indian Subcontinent has been very influential for most of history. In the modern era it has been under a repressive, explotative british government (which was established in large part due its disunity) and this has curtailed its influence in recent history. Hope that made sense

India has had an enormous influence over it's neighborhood historically. All of southeast Asia as well as parts west (Afghanistan, Iran, central Asia) were influenced by India historically. You only need to have some knowledge of Sanskrit and pick up any map of southeast Asia to recognize the huge number of Indian names still existing for dozens of towns and cities over Burma, Thailand, Cambodia, etc. Or pick up a history book and see how many of their kings and dynasties had Indian names.

Global influence is a different matter - no country had it until recently with the voyages of exploration, and more so with European industrialization. This was a down period in Indian history (it's had many ups and downs over 5,000 years), when the long wars between the Mughal Empire and the Maratha kingdoms drained both of wealth and power, and they were breaking up.

The "modern" world is very recent. The modern world has seen India come out of over a century of colonialism, plunge into the famines of the 50's and 60's when the monsoons failed, three major wars, global politics (where it was ostracized and embargoed by the west for not joining the war against the communist bloc), the need to feed hundreds of millions, bring the industrial revolution to such a huge country and catch up in a few decades with what took the west a couple centuries, etc. It's not surprising it lags behind.

But history isn't over yet, so who knows what is to come.

I think what the OP intends to ask is based largely on specific global perspectives. For example, as a U.S. citizen I often hear global news talk about how we're competing economically with China, or about how Russia is doing something with or against American politics. You don't often hear India mentioned in headlines like these (although that's a bit selective). In that context, a better question might be "What have the historical relations between India and the United States been like?" Or "What important political, cultural, or economical influences has India had on the Western Hemisphere in the 20th century?" Can anyone answer either of those?