In case it's relevant to you or anyone else all these years later, you can accomplish this using the Generate Near Table tool.
Well, the Generate Near Table tool only calculates the airline distance between points, i.e. it only accomplishes parts of what is proposed in this idea. The idea here goes beyond that, i.e. to have a solver in the network analyst generating O-/D matrices simultanuously for
(1) shortest paths over a network dataset (in meters, miles or km), plus
(2) airline distance (in meters, miles or km), plus
(3) manhattan distance (in meters, miles or km)
between origins and destinations. The result of this solver would then be one O/D matrix with all three information calculated in one run. At the moment, for (1) we need to apply the OD cost matrix solver in the network analyst, for (2) we have to run the Generate Neae Tables tool and as regards (3) I am not sure whether this is possible at all in ArcGIS Pro at the moment. In any case, sometimes there is need to compare results of all three indicators. Instead of happy to apply two or three individual tools it would be nice if one tool would allow for a simultanuous calculation. This would simplify (and thereby speed up) workflows significatly.
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