I don't understand your explanation. Perhaps I am inferring something you didn't mean.
The "it" in "It rained last night" is merely a placeholder, but it seems to me that the "it" in "I saw it happen" is a pronoun and must have an antecedent, which would be some previously introduced event, like "Jim ate his lunch" or "Someone sideswiped your car".
If you asked how I knew your car had been sideswiped, I might reply "I saw it happen". Here "it" clearly stands for a specific event, the sideswiping of your car. I agree that "it" gets its (accusative) case from "saw", which in the English syntax class I took would be analyzed as
(past) + "see" --> "saw".
Syntactically there are only two tenses in English: past and present. Only the first verb in a compound verb is inflected. The idea of future is implied periphrastically. Indicative mood in active voice starts with a sequence like the following
noun tense {modal} {have + -ed} {be + -ing} verb
which undergoes a transformation to produce what older grammarians called a tense, e.g., I eat (present), I am eating (present progressive), I have eaten (present perfect), I have been eating, I will eat (future), I would eat, I had eaten (past perfect), etc. Here tense, -ed, or -ing applies to the following verb. Syntactically and historically, the past tense of "will" is "would", of "shall" is "should", and similarly for the other modals. Curly braces surround optional elements.
The trouble with Noam Chomsky is that he keeps changing his mind. His ideas are a moving target.
I agree that "happen" doesn't have a tense in my example. After that, I get confused. Maybe "saw happen" is a compound verb, or maybe "it happen" is a clause. I'm not sure about this.