Memes

Other people have answered more thoroughly, but it should be added that even your source never calls grass fed beef carbon neutral (on the very first paragraph it even says that it isn't), just that it has a better CO2 footprint than grain fed beef (and that not by much, as has been pointed out)

It's not carbon neutral if you look at studies that account for more factors. For instance, here's an article with an interview of the researchers in the field talking about how there is no carbon-neutral beef

There’s not been a single study to say that we can have carbon-neutral beef

[...]

We also have to ask how much of the sequestered carbon in these systems is actually due to the cattle. What would happen to the land if it were simply left fallow?

The answer is, depending on the land, and on the kind of grazing, it might sequester even more carbon https://www.washingtonpost.com/food/2022/10/03/beef-soil-carbon-sequestration/

If we look at much more rigorous reviews on the carbon sequestration potential of "regenerative grazing" it's pretty slim. It cannot even sequester enough to counteract just grazing only production which only produces 1g protein/person/day

Ruminants in grazing-only systems emit about 1.32 Gt [...] These are their emissions. The question is, could grazing ruminants also help sequester carbon in soils, and if so to what extent might this compensate? As the following numbers show, the answer is ‘not much’. Global (as opposed to regional or per hectare) assessments of the sequestration potential through grassland management are actually few and far between, but range from about 0.3-0.8 Gt CO 2/yr 301,302,303 with the higher end estimate assuming a strong level of ambition.

https://www.oxfordmartin.ox.ac.uk/downloads/reports/fcrn_gnc_report.pdf

And keep in mind that this doesn't scale very well due to the massive land it requires. Already clearing land for pastures is a large deforester. Trying to even scale to a quarter of beef demand would require using 100% of grassland which would put enormous pressure for further deforestation

We model a nationwide transition [in the US] from grain- to grass-finishing systems using demographics of present-day beef cattle. In order to produce the same quantity of beef as the present-day system, we find that a nationwide shift to exclusively grass-fed beef would require increasing the national cattle herd from 77 to 100 million cattle, an increase of 30%. We also find that the current pastureland grass resource can support only 27% of the current beef supply (27 million cattle), an amount 30% smaller than prior estimates

[…]

If beef consumption is not reduced and is instead satisfied by greater imports of grass-fed beef, a switch to purely grass-fed systems would likely result in higher environmental costs, including higher overall methane emissions. Thus, only reductions in beef consumption can guarantee reductions in the environmental impact of US food systems.

https://iopscience.iop.org/article/10.1088/1748-9326/aad401

Are the majority of livestock grass fed?

I looked on google scholar to see any of their papers. Most of what I could find doesn't seem to be exactly peer-reviewed articles nor are most really cited by any other papers. At that they are making rather bold claims with rather weak evidence