Junk science. They just scooped up some dirt and did radiocarbon dating. No effort to make sure the sample was in context (e.g. from a layer above construction activity, or a cooking fire or something). So some dirt in the vicinity of the pyramid is 20k years old. That says absolutely nothing of value.
Unit 1 (surface): 2,000-1000BC Unit 2 : 6,000-5000 BC Unit 3: 24,000-14,000BC Unit 4: NO data
Unit 2 is clearly artificial and represents a construction layer. It has pieces of cut columnar basalt aligned parallel to the ground -- which does not occur naturally, so it is undisputedly artificial.
The soil filled on top of Unit 2 contains homogenous radiocarbon dates and no weathering indicating it was placed all at the same time.
They unit 3 layer contains weathered rounded boulders stacked on top of each other -- this type of weathering normally occurs in a stream an their presence on top of a hill does seem to indicate they were placed there. They also seem to be placed into structures forming a wall. While it is less obviously artificial than unit 2 -- they still provide some compelling evidence that it is indeed artificial.
They claim unit 4 is a natural andesite lava flow that has been "sculpted", but I don't see any evidence to back up that particular claim in the paper, unless i missed it.
TLDR: they did exactly what you mentioned -- took samples from above / below / inside various construction layers, and the dates match the stratigraphic context.... It wasn't just 'scooped up in the vicinity of the pyramid.
I agree given the nature of their claims, it would be ideal if they included another dating method like OSL to go along with the radiocarbon dates, but the radiocarbon dates themselves are not as half-hazard as you make them out to be.