No, in this situation, you definitely do not want to use had instead of would have. The change in meaning would be drastic. That icon they're talking about, at the time, did not yet have an incalculable influence on American labor and society. He or she was going to have that influence! This type of grammar is called the future in the past—we're talking about events that are going to happen in the future in terms of the past. Here's a quote from the link I posted above:
Future in the past is used to express the idea that in the past you thought something would happen in the future. It does not matter if you are correct or not. Future in the past follows the same basic rules as the simple future. "Would" is used to volunteer or promise, and "was going to" is used to plan. Moreover, both forms can be used to make predictions about the future.
You can always substitute forms like would have with was/were going to have and the meaning basically wouldn't change at all. For example:
I told you that I would do it and I did it. / I told you that I was going to do it and I did it.
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