Just like there was no Bronze age, Middle Ages or Third World. It doesn't mean that we cannot use the terminology or that it's necessarily wrong, it's just limited and we need to understand the limits
They did exist the same way people and goods did travel across Eurasia, but it wasn't some defined political or economical entity and they did not call themselves "the third world"
Unlike "the Silk Road" or "the Middle Ages", "third world" was a term in contemporary use at the time it applies to, including among non-aligned states and NGOs that worked on third world cooperation.
It was pretty well-defined as political classifications go, and people involved in actual "entities" related to it were aware of and sometimes used the term.
When originally coined (circa 1950 around the Korean War), the First World was the US aligned block of countries, the Second World was the USSR aligned block of countries, and the Third World was all of the countries not part of either. Egypt, India, Yugoslavia, Ghana and Indonesia viewed themselves as leaders of the broader political movement during the 1960's and 1970's.
Even into the 1960's there were few industrialized nations outside of those two main blocks, so "Third World" quickly lost its explicitly political meaning and became more a description of the level of capital investment and worker productivity.
ya, there were a bunch of trade routes along this path to all the different regions/cities. We just named the entire concept the "Silk Road" in the 1800s (it was coined in 1877 by Ferdinand von Richthofen).
"While engaged in a survey of China, the baron was charged with dreaming up a route for a railway linking Berlin to Beijing. This he named die Seidenstrassen, the Silk Roads. It was not until 1938 that the term Silk Road appeared in English, as the title of a popular book by a Nazi-sympathising Swedish explorer, Sven Hedin."
This just says that no one was aware the Silk Road existed, much less had named it that.
But it did exist, and goods were shipped between Europe and China without either side being aware the other existed, which at least I think is pretty darn amazing!