I strongly believe that the majority of people seek this, some earlier than others.
This is just a reduced form of below, correct?
I think you may be taking the concept of reduction too far. Grammatical reduction, in one sense, is a stylistic device that writers use the make sentences clearer and more concise. In another sense, it is a way of analyzing sentences for correctness. In your example, I doubt that anyone would have written or spoken this sentence and then reduced it. Most reduction takes place in the head, not on paper. Maybe you should say that the sentence below contains a “swollen” phrase.
So, it cannot be labeled a phrase in the reduced form, correct?
It’s a phrase. What else would you call a meaningful group of words that isn’t a clause?
The reduced from is better for both formal and informal writing, correct?
It’s different, not better. Everything depends on how a piece of writing hangs together stylistically. If you took the Magna Carta, The Sermon on the Mount, or the Declaration of Independence and reduced them grammatically, would they be better?
I strongly believe that the majority of people seek mind-altering substances, some (people seek this) earlier than others.
Glad you reduced this. Otherwise, you would have a comma splice. Also, this should be these.