How much does the productivity of modern agriculture depend on Glyphosate?
Chemistry is not a bad thing; life depends on it. And just because a pesticide or herbicide is natural or synthetic has practically nothing to do with its toxicity. Many so-called natural or organic pesticides are many times more potent than widely-used synthetic ones, as this article mentions.As to whether or not Glyphosate is necessary, this article explains some of the economic impacts from an outright Glyphosate ban. Some of the effects include:...French research institute Arvalis suggested a ban would cost that country’s agriculture industry 976 million euros ($1.1 billion). And in Germany, the Kleffmann Group said restrictions could hurt barley and corn output—with profit margin shrinkage between 40 and 70 percent–as farmers start planting alternative, more profitable crops.

Meanwhile, in the United Kingdom, where in 2014 2.2 million hectares of farms—a third of arable land—was treated with glyphosate, Oxford Economics estimated the impact would reduce wheat yields by 12 percent, oilseed/rape yields by 14 percent, and cereal production by 15 percent. Agricultural contributions to the nation’s GDP would shrink by about £930 million ($1.23 billion).In another article I recall reading (but can't find the citation for), Glyphosate is applied at a rate of only one pint per acre. Farms are not drenched in it as many anti-Glyphosate advocates would have us believe.As to whether or not Glyphosate is a carcinogen, this article calls much of that into question. Apparently, there were individuals involved who might have benefited substantially by having Glyphosate declared a human carcinogen.Edit:Some other interesting articles:This article  from Factcheck.org had this interesting tidbitQ: Does the herbicide glyphosate cause cancer? 

A: There is evidence to suggest it may cause cancer at very high doses, but not at the low doses typically found in foods.and
The European Chemicals Agency...concluded in March 2017 that “the available scientific evidence did not meet the criteria to classify glyphosate as a carcinogen.”And this article from the-scientist.com said A new study has found no conclusive link between exposure to glyphosate—the main ingredient in a popular weedkiller—and cancer.And Reuters in this piece  said:
The World Health Organization's cancer agency says a common weedkiller is "probably carcinogenic." The scientist leading that review knew of fresh data showing no cancer link - but he never mentioned it and the agency did not take it into account.