You are right insofar as rote memorization not being an ideal way to become a fluent language user, but "language acquisition model" is not a theoretical framework. Language Acquisition is a sub-field of linguistics.

"Comprehensible input" is an untestable hypothesis from the 1970s by a researcher named Krashen. Immersion methods are perfectly fine ways to acquire language--both grammar and vocabulary--but a massive benefit to already having a first language means that you can leverage your existing linguistic schemata (e.g., mappings for abstract concepts onto words, grammatical categories, etc.) to jumpstart your second language competencies.

With structured instruction and ample opportunities to practice speaking conversationally, a classroom learner can achieve the same level of conversational fluency as someone who learned the language immersively.

Further, a purely conversational course would not lead directly to improvement in the domains of reading and writing. There are some synergies, but these are separate skills that need to be targeted by specific pedagogic interventions. This is why children learning their first language still need to go to school to learn how to read, of course. And a major benefit of learning to read is then reading to learn.

The primary issue here is classroom time. Language instructors need to focus on a million different things. Here's a no comprehensive list off the top of my head: the domains of reading, writing, speaking, and listening; compositional modality (e.g. presentational speech, colloquial speech, presentational writing, genre-specific conventions for persuasiveness/humor/storytelling/etc.); general vocabulary and grammar; specific vocabulary and grammar (e.g. for home/academic/professional/etc domains); social norms (again by domain); cultural literacy (again by domain); etc.

And then divide the instructor's time by the number of students.

A learner needs to integrate within a speech community and continue practicing these skills within the appropriate contexts, or they atrophy. The foreign language context (i.e., the target language is not commonly spoken in daily life near the learner) is terrible at this, because it means that the learner does not have easy access to others with whom to practice and from whom to learn.

Tldr; use your other languages to help you speed up the baseline memorization and pattern recognition skills that are fundamental to contextual application, find a community, and do language with them.

My bona fides are a PhD in the subject and a decade of language teaching in US public schools and universities