Actually, as someone who does LCAs for a living and works with a lot with grass farmers (in the US northeast), the claims made in the article seem quite reasonable.
The carbon footprint of building and maintaining an aircraft is large, but that total footprint is spread across thousands of individual flights so when you're looking at a single flight the carbon footprint ends up quite small, most likely negligible.
Also, while in an ideal grass farming enterprise the grass does sequester carbon, I suspect that for the average grass-based enterprise this isn't true. The average grass farm, whether here in the US or across the pond in Europe, is poorly run and pastures most likely overgrazed. These farms probably don't sequester much carbon, and could very well be a net source of C to the atmosphere from the soil. And if you graze animals that have for generations been heavily grain fed, their genetics and microbiomes are not well suited to a diet of 100 percent grass so they do fart more methane.
While I think it's always useful to carry a healthy degree of skepticism when reading articles like the one RF linked to, I also think it's important not to dismiss then out of hand when they disagree with our closely held beliefs. Sometimes our beliefs aren't based on reality.