Swarmkeeper
Hero
I can see why you disagree, but if you rule it that way it is house-rule.
RAW, the Shove special attack can only be done in two ways: as part of "Using an Attack action" as written under the Shoving A Creature (as noted before); and second as a feature of the Shield Master feat, which as has been discussed ad nauseam and also requires the use of an Attack action. Since you can ONLY take Attack actions on your turn, you cannot Shove a Creature outside of your turn, i.e. as an OA reaction.
You are reading "melee attack" as "Attack action", they are two distinct, although similar, things. Again, if it doesn't make sense to follow the RAW, just change it for your game. No harm done if you want to play it that way.
I literally said sword swings and Shoves "both use the Attack Action, both are melee attacks, and both are able to be used as a Reaction". I did not read or say Attack Action = melee attack. My point: if a sword swing uses the Attack Action, is a melee attack, and can be used with a Reaction; AND a shove has the exact same 3 characteristics as the sword swing that I just outlined; then both can be used as a Reaction for an Opportunity Attack.
I think my interpretation is RAW, you think your interpretation is RAW. Both think the other is wrong. But perhaps we're actually both right because, like @DEFCON 1 says above: Rulings not Rules... only I'm righter because I like to give my players more "Yes"es than "No"s.
