  In recent work, Dao and Eisenbud define the notion of a Burch index,
expanding the notion of Burch rings of Dao, Kobayashi, and Takahashi, and show
that for any module over a ring of Burch index at least 2, its $n$th syzygy
contains direct summands of the residue field for $n=4$ or $5$ and all $n\geq
7$. We investigate how this behavior is explained by the bar resolution formed
from appropriate differential graded (dg) resolutions, yielding a new proof
that includes all $n\geq 5$, which is sharp. When the module is Golod, we use
instead the bar resolution formed from $A_\infty$ resolutions to identify such
$k$ summands explicitly for all $n\geq 4$ and show that the number of these
grows exponentially as the homological degree increases.
