I have no hard evidence to support this answer, but I guess that is somewhat inevitable as the question itself feels soft.
But it is nevertheless a perfectly valid question, and I hope this answer can at least give some food for thought.
On the other hand, medieval or Renaissance texts seem to have a word order that is more similar to modern Romance languages.
This is no surprise at all.
Latin and her daughters have evolved over time like any language.
Later forms of Latin are closer to modern Romance languages because Romance languages were not born from Plautine Latin.
It is also very likely that even after the emergence of Romance languages they and Latin developed in a somewhat similar direction.
They were (to some extent) used by the same communities and under the same pressures.
Simply put: Things change over time, so the further you go back into the past, the further you can expect things to be from their modern day form as a rule of thumb.
Whenever I see Classical texts, the syntax of almost every sentence is really complex.
My impression is that a sentence is a different kind of semantic unit in English and Latin.
The English concept of a sentence is pretty clear: it is indicated with capitalization and punctuation without much ambiguity.
Classical Latin used almost no punctuation at all.
Splitting ancient texts to sentences was a later editorial process.
In English and many other modern languages sentence boundaries are clear and hard.
Perhaps the Romans did not feel the need to fully stop one sentence before starting the next one, but allowed sentences to fade into each other.
When you try to punctuate that without changing word order so that you conform to modern definitions of a sentence, you end up with horribly long sentences.
Many translations split Latin sentences into shorter ones.
This shortening could often be done to the Latin originals, but this has probably been considered as too much of an injustice towards the originals.
I am by no means sure that this is what actually happened, but this way it seems to make sense to me.
I think the Romans simply had a different view to structuring language, and perhaps sentence was not as crucial a concept as it is to us.
It would be very interesting to hear how the Romans understood a sentence if there are any extant accounts.
It is also good to remember that there is a genre bias.
We do not have short newspaper articles, shopping lists, or tweets from the antiquity.
The language that has been conserved consists of high prose, which is not identical to every day language in any culture I know.
How could people of Ancient Rome read those texts with fluency?
The Latin that has survived to us from antiquity is the language of the elite.
Literacy levels were not where they are now, and there has been a significant bias in choosing what texts to save.
Most of the Latin produced in the classical era is lost.
Inscriptions and other archaeologically found texts (as opposed to those preserved by two millennia of philological tradition) may be more "honest" in preserving everything equally, but written text in any form was not in everyone's reach.
And those texts are far shorter than the usual kind of Latin literature, not leaving room for the kind of elaborate sentences you will encounter in Cicero or Livy.
Most people of Ancient Rome could not read those texts with fluency.
I would assume written language to develop quite differently from the modern world when used by only a small fraction of the population.
So, were Classical authors sophisticated when writing just because that style was really natural and easy to understand, or was it just for the sake of elegance and for showing off that they were part of an elite?
It would probably have been more efficient to communicate in shorter and simpler sentences, but literature is not only communication but also art.
The authors and their audience were well educated, and would probably appreciate elaborate structures more than the average modern reader.
The artistic preferences are different when the art form is shared by the entire population.
I don't always choose the simplest possible way to express myself.
I enjoy using more sophisticated and nuanced language when possible.
So did the Romans, but to varying extents.
If you read different authors, you will see big differences in syntactical complexity.
The authors had to appeal to the taste of the elite, as there was nobody else to read their work.