My methods were pretty much the same as the contractor shown in the video. I used a 4' fiberglass probe. Note that I started and learned in NC, so not too surprised at this contractors methods. IMHO, the NC code provides better guidance for prescriptive construction in this area. Note that the tables he shows in his NC code are underlined, which means that those sections are amended from the ICC code and are specific to the NC code. That said, like many others who have contributed, the probe only came out when concerns presented themselves, and only as a way to demonstrate the validity of my concerns. This was rare in the areas I worked in NC, but later, in another state, the probe got more work. I would simply state that I had concerns about the soil bearing capacity and why, and would take pictures of the buried probe for use when questioned about my concerns by the boss. I also occasionally used a penetrometer, but found that to be less impactful than the pictures of the probe buried to the hilt. I didn't recommend any particular measure to mitigate the questionable soil beyond find better soil or get an EJ. Can't recall, in any state, an engineer accepting any of the soils I questioned without some serious mitigation efforts.
Also, note than in NC, there was a mechanism to appeal a building officials conclusions to the state board, at which point they would investigate through their department of engineering. Buried probes would go a long way in those cases.
Where I am now, most AHJ's require a geotechnical report and an engineered foundation, and due to liability, nobody complains. My probe is pretty much retired.