Fine.
Remember.. and this is key. The definition of an opportunity attack does not say "sword swing, etc." It says "melee attack". Sword Swings, shoves and grapples all fall under melee attack. They are also all defined as "attack actions" (which makes it irrelevant when discussing OA). Therefore... OA away.
Oy...
Even though they all fall under "Melee attack", and swinging a sword (and other such typical forms of attacking), OA, TWF, Grapple, and Shove are all types of melee attack, however those with sub-sections have special conditions to allow their use. That is why they
have sub-sections to deal with those specifics or special conditions. Without those conditions, you can't do them. Why are you ignoring those conditions?
No reaction available? No OA.
No bonus action available? No TWF.
Not using the Attack action? No TWF, no Grapple, no Shove.
I mean, it is right there in front of you for grappling:
"When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple."
Shoving is even more obvious:
"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."
Do you want to grab a creature or shove it? Yes.
How do you grab a creature? I can use the Attack action to make a special melee attack, known as a grapple.
How do you shove a creature? Using the Attack action, I can make a special melee attack to shove a creature, blah blah blah.
Do you get the idea that the word "can" is used to tell you these are options you get to employ if you are using the Attack action?
Also, where else,
anywhere, does it say under any of the other sub-sections of melee attack or main melee attack section that you can make any kind of melee attack into a grapple?
Player: "I want to grapple that creature since it's moving out of my reach."
DM: "You want to grapple?"
Player: "Uh, yeah, that's what I said."
DM: "Ok, can you use the Attack action?"
Player: "Well, no, it isn't my turn. I get an opportunity attack when it moves, though."
DM: "Sure, you get the opportunity attack, but without the Attack action to make the special melee attack, you know, the grapple, you can't do it now as a reaction."
Yes, OA provide a melee attack as a reaction. But it is as a reaction, and you cannot take an Attack action as a reaction, only as an action on your turn.
You are choosing to ignore specific rules over the general rules, and that is not the way the game works. You know (or I hope you do I suppose) that specific rules beat general. Monks don't have proficiency in longswords, but a High Elf always does, so a High Elf Monk could use a longsword with proficiency.
Consider this: a Monk is attacking with his longsword. The DM knows Monks don't have proficiency in longswords and tells the player not to add my proficiency bonus to the attack roll. The player reminds him the monk is a High Elf, and has proficiency with longswords, thus allowing him to add his proficiency bonus to the attack. Later on the Dwarf Wizard purchases a longsword and uses it to attack with. The DM again, informs that player not to add his proficiency bonus to the attack roll. The player responds, "But the Monk could do it." The DM replies that was because the other character is a High Elf and has proficiency in longswords due to his race, thus meeting the requirements to add his proficiency bonus. Dwarves don't have that feature so the player with the Dwarf doesn't get to add the proficiency bonus to the attack roll.
You are assuming that since you can shove or grapple under one circumstance, you should be able to do it under any. That is not the official rules. Making an OA does not allow you an Attack action, it only grants a melee attack as a reaction. Although a shove or grapple is a type of melee attack, they have conditions requiring the use of the Attack action.
As I wrote in the other post, I think at the DMs discretion, he could allow grapples or shoves as OAs under the right circumstances, but he is breaking from the official rules and is completely within his right to do so.