Why doesn't this reaction happen? In the synthesis of DMAD, we eliminate two bromides from dibromosuccinic acid in basic environment to get acetylenedicarboxlyic acid (Step 2).


However, this reaction could also happen:


The reaction in Organic Syntheses (http://www.orgsyn.org/demo.aspx?prep=cv2p0010) went with a yield of 72-88 %, which means that these pathways probably do happen, but only in a minority of cases.

Why is that so, given that the elimination of CO2 is entropically favoured?
This elimination is unfavorable because C--C bonds are generally quite stable. Decarboxylation would involve the formation of an unstable carbanion before proceeding to the elimination as you have outlined. (Note, this is more favorable under different functional group conditions, namely when a neighboring carbonyl can stabilize the carbanion through the formation of an enol  https://www.masterorganicchemistry.com/tips/decarboxylation/ ) So, the formation of the alkyne is favored here over the elimination of CO2.