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Is it grammatically correct to put an "article" before "follow the rule"?
For example:
I know the second one is correct, but if possible, I'd like to simplify the sentence as much as I can.
Yes, it is absolutely correct to do so, as it refers to an existing Grammar rule and "the" is a reference back to that particular rule.
As for the word product, if you are referring to a particular product, which has already been mentioned; you may say, "the product" else "a product."
Having said that, the simple answer to your question is, an unequivocal, "Yes".
There is a subtle difference between the two sentences: the first says the product follows the rule, the second says making the product follows the rule (suggesting that the rule is about the manufacturing process).
It's also hard to say whether "the" is correct without a specific context. For example, the manufacturing process might "follow OSHA safety rules," or it might "follow the OSHA safety rules." The first gives a sense of following OSHA's rules instead of not following any rules; the second gives a sense of selecting OSHA's rules, instead of a different set of rules.
