Soap is traditionally made through a process known as «saponification» where a strong alkali, like sodium hydroxide (solid soap) or potassium hydroxide (liquid soap), is added to fat/oil. The result is glycerin and the salts of the fatty acids (soap).
The pH of soap is inherently alkaline because of the saponification process. Synthetic surfactants can also have a high pH. For example, my sodium coco sulfate has a pH of 9.5-12 in a 1% aqueous solution. However, the pH of soap cannot be adjusted in the same way that the pH of a synthetic surfactant can be. Lowering the pH of soap can cause re-acidification of the fatty acid salts and potentially cause the soap to separate or lose its effectiveness. It would also require a much higher addition of acid to significantly lower the pH which would likely increase the potential for irritation even if it didn’t compromise the structure of the soap.
Both soap and synthetic surfactants can be effective cleansers (some more than others), but traditional soap has a higher potential to leave a residue (soap scum) in hard water conditions than synthetic surfactants do.
I think at the end of the day it is largely a personal preference. If your skin is not irritated by traditional soap, then it should be completely fine to use. Traditional soap is still popular in many cultures. In Palestine, we have Nabulsi Soap, which is a 100% olive oil soap with both solid and liquid varieties that has been made since the 10th century and is still very popular and common in our country to this day.
I think it also depends on the usage. For example, using traditional soap as shampoo for your hair will probably never get you the same performance and results as a quality shampoo made with synthetic surfactants and other ingredients. But, for washing the hands, I have no problem using traditional soap.