Yes. And no. It depends on which more specific question you are asking.
So-called "Vulgar Latin" was indeed a spoken language, no question about it. There is historical evidence as well as the evidence shown by the modern languages today (that they come from a common ancestor).
Classical Latin however was not a spoken language. As you read it, it is a highly stylistic version of Latin used as a literary language. Some more formal or poetic speakers may have presented themselves in a manner similar to that but it would be like speaking formally/stylistically today, not average daily language.
As for "complexity", no, it is not more grammatically complex than various spoken languages. It is simply different from your expectations and probably simpler in some other ways you might not have noticed such as lacking articles (the, a). So stylistically, no, Latin never was used quite like what you see written. But grammatically, more or less, yes, people did speak in that "complex" (=different) way, and speakers of many languages still do today (for whatever specific grammatical feature you want to pick or, however you want to count them, about the same "number" of such features too). Try Russian for example if you find the cases confusing. Or some Australian languages with even more flexible word order (which can be also used stylistically but also naturally in many ways). Latin is not an outlier grammatically.