As someone relatively new to cast iron (around a year of cooking with it), I am inclined to believe the parts of this article that I cannot validate through my own experience, because the other parts match what I have learned through experimentation. However, I must point out that a year ago, I would have no way to recognize this. Whether or not this article is correct, it is at the same level of heresay as every other article, reddit comment, or youtube video out there (like the carbon steel video linked in the replies, which says never to use soap).
Despite the flax oil article being bad science, it is still the most science of any of the cast iron information out there, which is why it continues to enjoy such popularity. Even if it is bad science, it at least provides some hypotheses to test!
Here's my hypotheses on soap (edit: rephrased to better reflect my meaning):
1) Soap is fine to use; it will not damage the seasoning.
2) Seasoning is often damaged (at least a little) while cooking and it is important to re-season regularly to keep the pan in good shape.
3) If you do not use soap, the amount of oil left on the surface is similar to the amount you would want to wipe on when re-seasoning.
Basically, I think no soap + dry on stove is a shortcut to avoid the process/effort of reseasoning the pan (especially if you do the oil-and-salt-as-an-abrasive trick). Perhaps you'll get better results using soap and then wiping on oil, but it's more work and many people don't do the second part. Thus, they see better results when they don't use soap.
This would explain why some people tell you you need to reseason after every use, while others say that's bubkis, and why the same is true of soap avoidance.
To test: try all 4 combinations of soap/no-soap and reseasoning/just-drying (always dry the same way, on the stovetop).