> The first time it happened, as I understand, there simply wasn't a huge amount of oxygen in the air
When cyanobacteria (blue-green algae) originally cleaned out Earth's CO2 atmosphere and produced O2, I don't know where the C went. To bedrock as calcium carbonates (CaCO3)?
But the carbon that humankind has released into atmosphere by burning coal and oil, was sequestered by plants during the Carboniferous+Permian periods (360 to 250 million years ago). These were large, vascular plants. Not too much different form plants today. And the atmospheric oxygen was already at about current levels.
But your idea might hold in a different context:
There is a theory that 360My ago plants invented lignin, to make their stems more rigid and woody. And then it took time before fungi were able to invent enzymes that are able to digest lignin. So maybe the the large burial of biomass carbon in Carboniferous could not happen again, with out modern fungi being able to decompose wood.
And there is another theory, that as Carboniferous+Permian periods were during an ice age, the low sea levels allowed for large areas of swampy lowland vegetation. Where vegetation sequestered carbon, and wet swampy soil allowed the carbon to get slowly buried, because microbes cannot as easily decompose biomass in wet, anaerobic conditions.
https://en.wikipedia.org/wiki/Carboniferous#Rocks_and_coal
https://en.wikipedia.org/wiki/Coal#Formation