What does “universal or linear mechanism” mean?

Universal

A universal mechanism would mean that the same process applies to virtually everyone in the relevant population.

For example:

“Once former slaves became legally free, they became members of the middle class.”

That would be universal.

You are not saying that.

You are saying:

Former slaves became legally free → they could participate as free workers → some eventually improved their economic position sufficiently to enter the middle class.

That is not universal.

Linear

A linear mechanism means a process in which A leads straightforwardly to B, B to C, and C to D, with little possibility of interruption, variation, or different outcomes.

For example:

Freedom → factory employment → wealth → middle class.

Your argument is not quite that simple because you recognize the continuing importance of unequal assets and access to capital.

So when I said:

“The evidence does not support such a universal or linear mechanism,”

I was responding to a stronger proposition than the one you actually intend.

And yes, if we restrict the geographical scope to Europe and the United States, the question becomes narrower. The relevant question is not whether this happened everywhere in the world, but whether your proposed process is historically defensible within the European and U.S. contexts you are discussing.