https://prairiesoilsandcrops.ca/articles/volume-1-3-print.pd...
From same section of the article it was picked from :
"While increased use of grains in ruminant diets reduces enteric CH4 emissions, there is concern that increased grain production may increase the use of fossil fuels for fertilizer, machinery, and transport, resulting in more greenhouse gas emissions. Grain feeding ignores the importance of ruminants in converting fibrous feeds, unsuitable for human consumption, to high-quality protein sources (i.e. milk and meat). Furthermore, high grain diets can negatively affect cow health due to acidosis. With escalating grain prices, the scope of further increasing the grain content of ruminant diets in Canada is limited"
Even ignoring CO2 cost, nutrition, grain availability and all other complications, what is required to show that grass fed diets necessarily lead to more methane output than alternatives - is a comprehensive study of the performance of all dietary options and supplements. I dont think that is setting too high a bar, to avoid arguing on sweeping generalization and loss of context.
Taste can often be regarded as ephemeral, while fast fattened livestock can be discerned to taste different and are considered inferior in most food celebrating cultures.
There is a possible health factor involved with grass fed (or mixed prairie for better) beef and dairy accumulating a markedly different spectrum of omega oils, which are debated inconclusively, but also formally studied and theorized to be superior for human consumption.
A focus on the strength of methane emissions seems increasingly common in discussions and magazine articles, while the long developed advice from the IPCC is that CO2 demands priority because methane clears naturally in a decade or so, and requires less action to avoid than CO2 output which takes much longer to clear.
My understanding of IPCCs focus on CO2, is that while methane reduction presents an opportunity to buy a few years time, the priority is to convince action on the hardest problem which has been created, is worsening rapidly and much harder to clear.