This isn’t really true — in fact, it’s exactly the same fallacy as in Swift’s article. Certainly at a glance Greek and Latin
look a lot more complicated than English does, and certainly this is true with respect to their morphology (how words are created from smaller parts), but morphology is far from the only part of language. For instance: word order is pretty much entirely unconstrained in Latin, but English has much more rigid rules about word order, which are nonetheless often fairly subtle and complex to delineate. Or: we can distinguish no less than 11 different tense/aspect combinations in English, but this is non-obvious because they rely on combinations of verb form and auxiliary verb(s), and the auxiliaries themselves nearly all have several different interpretations depending on where and how they are used (see Dixon’s
A Semantic Approach to English Grammar for more).
Besides, even when you just look at morphology alone, ancient languages are no more complex on the whole than modern ones. Thus consider Russian, which has more cases than Latin (IIRC). Or Hindustani, which uses an entirely different case system depending on the aspect of the verb. Or Swahili, where the verb inflects for both direct subject and object (but not always!), which have no less than 18 irregular genders, as well as for 28 tense/aspect/mood forms (if Wikipedia is to be trusted) — amongst other categories.
What happens, it turns out, is that languages can grow more complex over time as well as simpler. People do tend naturally to simplify, delete and abbreviate syllables in words — but they also tend to merge words together and elaborate simple constructions to more complex ones, in order to retain the expressive capabilities of their language. I’ve already mentioned Deutscher’s bool The Unfolding of Language; I will recommend it again here, since it has a great summary of exactly how these processes work and interact with each other in order to generate complexity in language.