Opposed Strength/Dexterity Checks: does a Shove give you two for one?
From a game mechanics perspective, the question asks whether a creature can get a "two for one" outcome from the same opposed Strength/Dexterity check (a single die roll): does that one roll both accomplish the Shove and Escape from the Grapple action? This might run afoul of the action economy (which is slightly skewed in favor of the grappler).
The Grapple and the Shove are both considered a special melee attack, a subset of the Attack Action. Escaping a Grapple requires an action. (Basic Rules p. 74). Taking an action replaces using the attack action in most cases. Shove, Grapple, and Escape Grapple are opposed Strength/Dexterity checks.
The Rules Problem
The rules on Grappling and Escaping Grapples were not written symmetrically. Grapple is a special attack action, but breaking a grapple is not a special attack action, it is an action. (Note: a ruling could be to make the escape a special attack action as well, but that may not resolve the other issues involved).
Opposed Strength/Dexterity Check 1: Escape Grapple. (Basic Rules p. 74)
A grappled creature can use its action to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check contested by your Strength (Athletics) check.
Opposed Strength/Dextrity Check 2: Shove (Basic Rules p. 74)
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you’re able to make multiple attacks with the Attack action, this attack replaces one of them. The target must be no more than one size larger than you and must be within your reach. Instead of making an attack roll, you make a Strength (Athletics) check contested by the target’s Strength (Athletics) or Dexterity (Acrobatics) check (the target chooses the ability to use). If you win the contest, you either knock the target prone or push it 5 feet away from you.
Shoving a Creature.
If the distance between combatants is assumed to be some number greater than zero, then adding 5' to that makes the distance greater than 5' and the conditions for breaking the Grapple have been met for creatures with a reach of 5' or less. For creatures with a reach of 10' or more, it would not appear to suffice.
If the distance between grappled combatants is assumed to be zero, since they are in physical contact, then the shove by itself cannot generate enough space to break the grapple.
Recommended Ruling: Conservative.
Because the rules as they stand now demand slightly more to break the grapple -- requiring an action versus a special attack action -- then the opposed strength check necessary to break a grapple is the Escape Grapple Action.
Recommended Ruling: Permissive
Make the Grapple Escape the equivalent of Grapple or Shove, a special attack action, and fit it into the action economy. If the attendant ruling on distance is that the two creatures begin at a distance greater than 0 from each other you'd get the two for one feature as a byproduct of the rules overlapping.
Use a Shove to break a grapple as a special attack action in lieu of Escape Grapple (they are both opposed strength checks) so that it does not impose the same penalty on the graplee of using an action. (Creatures having multiple attack actions would be more able to exploit escaping a grapple). This could be substituted in for the "prone" option of a Shove. A house rule along these lines should be closer to a balanced action economy.