Consumption of NAD+ in glycolysis
NAD+ is important in this step, since it is co-factor for the glyceraldehyde-3-phosphate dehydrogenase (G3PDH), which acts as a acceptor for the hydrogen atom from the C1 (see below).



If you look at the reaction, the aldehyde from the C1 is oxidized to a carboxylic acid which in a second step is turned into a phosphoester. To do so, a cysteine from the active center of the G3PDH attacks the G3P and forms a hemiacetal. This is oxidized into a thioester, the hydride which is split off in this step is transfered to the NAD+ bound by the G3PDH. In the second step the thioester formed between the enzyme and the G3P is attacked by a molecule of inorganic phosphate to form 1,3-BPG. This step is only made possible by splitting the very energy rich thioester bond. So for this reaction to happen you need both steps, as the formation of an energy rich phosphate ester bond would otherwise not happen. See here for more details.

To go directly from G3P to 3-PG you would need an enzyme which oxidizes the aldehyde group in the G3P. And then you need again an acceptor for the hydrogen which is split off here.