This is how Player's Handbook (p. 194) describes what counts as an attack:
If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack
There are ambiguous exceptions from this rule though — so-called "special attacks". There are only two of them — Grapple and Shove from the PHB.
As a DM, for the sake of clarity and consistency I want to call Grapple just "an action" or "a contest", not "special melee attack". So instead of
you can use the Attack action to make a special melee attack, a grapple
the text of the house rule (or the errata proposal, if you wish) will be
you can use the Attack action to make an action in combat, a grapple
So does for Shove. This is also consistent with the PHB "Contests in Combat" (emphasis mine):
Battle often involves pitting your prowess against that of your foe. Such a challenge is represented by a contest. This section includes the most common contests that require an action in combat: grappling and shoving a creature. The DM can use these contests as models for improvising others.
So the lowercase "attack" is changed to "action" or "contest". For instance, the next passage in the Grappling description will be "If you're able to make multiple attacks with the Attack action, this contest replaces one of them". The rest of the rules remains unchanged.
I want to figure out what possible impact this will make to the game balance. For instance, will some feats/class features change substantially.
In terms of the game mechanics, what consequences/repercussions this change will have? For sanity's sake let's confine ourselves to the the Player's Handbook.