Because "have got" is an informal colloquialism, it's not about correctness as much as what sounds natural. The speaker is barely considering the tense of the sentence. Think of it as a simple rule of word replacement: the words "have/has got" can replace the main verb "have/has," but only when it has no auxiliaries. It also works where "have to" (pronounced "haff to" in many dialects") has replaced "must." From there, you can contract or drop the "have" entirely:
I've got a dollar.
I gotta go!
By the way, the past participle "gotten" is unrelated to all this.
Here's my guess why, in example sentences:
The nicest simple sentences have got exactly one auxiliary.
Using none produces a more sterile tone. Academic contexts may expect this, and robot overlords demand it.
But even though correct and sometimes necessary, soon all your extra helping verbs will have had sounded so awkward!
Sometimes people kinda use other fillers or constructions to help. They tend to avoid using a single lonely verb.
There are exceptions; linking verbs sound natural alone. Brief phrases pass too. Use command forms by themselves. Common idioms work like a charm. Even among friends, broad or meaningful topics transcend this. An occasional simple sentence flows naturally, but close friends just won't wanna do it all the time.
But I hope nobody takes me for an expert. I even verbify my own wordages sometimes!
I'm wondering if I've ever thought about this until now...
