“Smith cannot carry the entire burden”

Here burden means:

the amount of evidentiary work a source must do to establish your claim.

Suppose your claim has four components:

1. emancipation created legal freedom;
2. racial attitudes survived emancipation;
3. formerly enslaved people were not immediately treated as social equals;
4. unequal land and economic resources constrained their freedom.

It would be a mistake to find one Smith article and cite it after all four statements.

Smith might establish #1 conceptually or #2 conceptually, while Kolchin, Perrone, and other historians establish the historical evidence for #3 and #4.

So “don't make Smith carry the entire burden” means:

Don't ask one source to establish propositions that actually require several different kinds of evidence.