I am bit confused with structure of this sentence, how the subject "Another American" is connected with the verb "claiming" in this sentence. Here it reads "Another American scholar chided me for being too obsessed with the electoral and the populist claiming that the roots of a moral, vital secularism have to be sought in history.

Look at the sentence this way:

"Another American scholar chided me for being too obsessed with the electoral and the populist, who claim / , and their claiming that the roots of a moral, vital secularism have to be sought in history."