Rotting grass produces the same amount of methane as cows that digest grass (see this source). So if cows did not eat the grass, it would still emit methane. Therefore, if there were fewer cows, would the total methane emission change at all?
Couple of thoughts:
(The OP's article is a low quality source of info with either an interview taken out of context or an inappropriately vague or under-informed interviewee. Reasons why will become apparent throughout my answer.)
Much of the world's grasslands are maintained for and by grazing livestock. Without dedicating so much of Earth's surface to livestock, we would almost certainly have less grass.
Though, much of the world's livestock are fed corn vs non-cereal grasses more commonly referred simply to as "grass". It's unclear if the OP's question is referring to grasses taxonomically (i.e.,inclusive of cereal crops) or in a general sense regarding grass leaf blades only. Regardless, my initial claim applies to presence of both pastoralized grasslands and croplands. The distinction is more important for a later point I'll make below.
See here for more info.
Grass does not always "rot." Methanogenesis is an anaerobic process. The bacteria in a ruminant's stomach are anaerobic, as are bacteria in various wetlands (or in landfills). However, the large majority of grasses do not grow nor die in these anaerobic conditions, so methanogenic bacteria would not be responsible for breaking down most grass. Instead, most decomposition produces carbon dioxide. See here for more info.
Relative abundance of various bacterial types in a ruminant's gut is dependent on that particular livestock's diet. Methanogens (i.e., methane-producing bacteria) only become dominant when the cow consumes a high-sugar feed (vs. a high cellulosic feed) -- e.g., corn vs other grasses. See here for more info.
A quick review of the methane cycle might also prove beneficial. Methane is broken down both by UV light and by methanotrophs (though the latter is more important for preventing CH4 from entering the atmosphere -- e.g., in places like wetlands -- and the former is more important for breaking down existing atmospheric methane). Folland et al. (2001) estimated that the "lifespan" of methane in the atmosphere is ~9.6 years. As such, the rate of methane production is important, as slower production will result in less accumulation. This is important when one considers that decomposition of plant matter in wetlands, for example (though quite variable based on conditions) can take years (vs hours/days in a ruminants stomach). From EPA:
Decomposition of emergent macrophytes in lacustrine wetlands may take from about 200 to 1000 days for 90 percent weight loss (Hill 1985). Breakdown rates (per day) range from 0.0008 for woody plants in bogs to 0.0190 for non-woody plants in riparian wetlands (Webster and Benfield 1986)
I'll try to update later with more thoughts, but this is a good enough starting point to post.
Folland, C.K., T.R. Karl, J.R. Christy, R.A. Clarke, G.V. Gruza, J. Jouzel, M.E. Mann, J. Oerlemans, M.J. Salinger and S.-W. Wang, 2001: Observed Climate Variability and Change. In: J.T. Houghton, Y. Ding, D.J. Griggs, M. Noguer, P.J. van der Linden, X. Dai, K. Maskell, and C.A. Johnson (eds.). Climate Change 2001: The Scientific Basis. Contribution of Working Group I to the Third Assessment Report of the Intergovernmental Panel on Climate Change. Cambridge, UK, and New York, NY: Cambridge University Press.