I read the goal as "to retrieve the values via both the number index corresponding to their postions and the meaningful keys"

Retrieving a subexpression via index is what Part (the double brackets) does.

But, if you want "uniform access" exactly, maybe you could use Dataset:

BasicVectorsData = Dataset[Take[BasicVectors, 14]];

Now you have:

BasicVectorsData[1] == BasicVectorsData["TricPrim"]

Why is that a problem?

BasicVectors["TricPrim"] == BasicVectors[[1]]
(*True*)

What this shows is that you can access an Association with Part.