https://www.lingvolive.com/ru-ru/translate/ru-en/аргументировать
По-моему, argue - это скорее "спорить", чем "аргументировать"

Согласен - даже не могу представить в каком контексте можно применить argue в значении аргументировать...
to give arguments/ground for, to prove - это гораздо правдоподобнее.
https://idioms.thefreedictionary.com/make+one%27s+point

Достаточно часто использую в написании сочинений при подготовке к ielts.. вполне уместно..

Я нашему американскому другу запрос послал. Жду ответа и поделюсь. Или он сам свой коммент сюда напишет.

There are several ways of expressing this. The most obvious way is using the verb ARGUE as a transitive verb: "He argued his position convincingly," which basically is another way of saying that he defended his opinion on a certain matter. (When ARGUE is used as an intransitive verb, it means to engage in an aggressive confrontation: They argued for two hours.) In a more formal manner, one could say that "he proffered a convincing argument for his position." So in my opinion, ARGUE is the number one answer.

As for the other answers, I've never heard GROUND used like this. SUBSTANTIATE refers to the result, whereas I believe АРГУМЕНТИРОВАТЬ refers to the process. Similarly, MAKING ONE'S POINT refers to the result. Correct me if I'm wrong.

Thanks Uly. And what about to make one’s point?

Oh, got it. So, he made his point means that his point was accepted, right?

Аргументировать - это процесс, именно так.

...and he made his point = he managed to prove he was right. Correct?

Correctomundo!

Understoondo!

👍