The inexorable expansion of the federal government and decline of relative importance of states followed the WBTS (though not for some decades), therefore it was caused by the WBTS.
Let us posit a world in which Africa was missing, and so no Africans had been imported to America. No slavery, no WBTS.
America would still have been settled by Europeans, and eventually probably have broken away from the mother country. Industrialization and its discontents would still have grown up during the later 19th and early 20th centuries, with resultant calls for government action to address those discontents.
In America the movement to do so was called the Progressive movement. But the exact same calls for expansion of central government power occurred in every other country that reached a similar stage in its development. Precise political issues varied by country, but the general trend is clear.
To believe that Lincoln "caused" the growth of federal power is just silly, IMO. He provided a significant precedent for such expansion, but the notion that TR, Wilson, etc. would have been unable to advance their notions without such a precedent is not logical.
Most of the federal growth during the war went away for the next few decades, not starting up again till the Progressives got going. AFAIK, they seldom cited Lincoln or the War as justification for the proposals.
Absent Africa there were still slaves, slaves existed long before anyone in Europe met an African.
Quite possibly the war and the experience of troubles afterwards, Reconstruction and the like made people more cautious and suspicious of federal power too, maybe why our leftists were less successful for a long time