The thing is... the Romans did advance quite a bit in engineering. They were good at solving practical and huge scale problems as their aqueducts, roads, amphitheatres, etc, manifest.
The thing the Romans didn't have was a scientific tradition. The Grecorroman world still saw the ancient Greek philosophers as the source of all knowledge, there was no need to investigate the physical world because after all, the old philosophers explained everything, and what was TRULY important was Philosophy on itself, and ethics, metaphysics... there just wasn't an idea of going on and doing experiments and observations as scientists did. And when Christianity came, the focus was shifted into theology, just look at the endless Christological debates, they sound confusing and even a bit ridiculous for us, but at the time, they were very important discussions about the Truth of things, and to set Christianity within a logical framework (inherited again from the Greek philosophers) something vitally important.
At the time, wise men (and a few women) weren't what we would call scientists or naturalists or engineers; they were philosophers and theologists. We know a lot about the philosophers and theologists of the time. We know little about who built the Roman Aqueducts, or the Antikythera mechanism. Evidently, they just weren't considered as important.
There were exceptions of course; the most famous one was Pliny the Elder who was perhaps the first naturalist and who made Naturalis Historia which was the go-to book for centuries with his word taken as fact (see a pattern here) and the Library of Alexandria was still active, if not at the heights of the Hellenistic period. But still, even with that, the Roman Empire lacked the scientific revolution that shaped the world from the Renaissance onwards.
I don't really think slavery has much to do with their technological stagnation. The Greek world also had its fair share of slaves, which did not stop scientific advancement.
Carl Sagan in Cosmos (the book) proposes that the advances of Greek proto-science were stopped because the Greek philosophers decided to concentrate in the metaphysical realm instead of doing experimentation and observation in reality which is vital to science, and he argues that this was the case because the overreliance of the Greeks on slaves made any kind of physical labor -and so experimentation- undesirable and even beneath the role of a philosopher, so eventually any possible scientific revolution was nipped in the bud and "wasted" in metaphysical, quasi-religious speculation.
I don't know how true this is, and certainly Carl Sagan was no historian, but I've seen the claim numerous times.