You could not perform that series of actions.

The reason for this is you only have the option to use only one bonus action and one action on a turn (barring certain exceptions like Action Surge or Haste).

• If you take the Dash action as your action for the turn, then you could make the bonus action attack with the Charger feat but you will have already used your action to Dash so you cannot take the Attack action.
• If you use Cunning Action to Dash, you will have already used your bonus action so you cannot make the Charger attack, but you could then use the Attack action.

The bonus action attack from Charger does not count as an Attack action

See this question for clarity on the difference between an attack and the Attack action

Charge as a regular action

You found what there is to find: "Charge" is not a regular action you can take.

The common conception seems to be that a player character is not big enough to charge something efficiently. In fact, most (to not say all - I couldn't find any that wasn't at least Large, but I don't like saying "all") of the beasts with the Charge feature are large beasts, while player characters are only medium. Example of Charge feature from the Elk:

Charge. If the Elk moves at least 20 feet straight towards a target and then hits it with a ram attack on the same turn, the target takes an extra 7 (2d6) damage. If the target is a creature, it must succeed on a DC 13 Strength saving throw or be knocked prone.

The Cavalier - Ferocious Charger

Additional to the Charger feat, the Cavalier subclass of Fighter, described in Xanathar's Guide to Everything, p. 30 and 31, has the Ferocious Charger feature:

Starting at 15th level, you can run down your foes, whether you’re mounted or not. If you move at least 10 feet in a straight line right before attacking a creature and you hit it with the attack, that target must succeed on a Strength saving throw (DC 8 + your proficiency bonus + your Strength modifier) or be knocked prone. You can use this feature only once on each of your turns.

This is probably the closest you can get to the Charge feature from beasts, although it does not add extra damage and only comes by 15th level, so you might be disappointed.

So, TL;DR: Charge is a special feature from a specific class, or you could take the Charger feat, but it is not a regular action anyone could take.

The Move + Shove scenario

As a side note, I would like to address this statement:

In other words, the player's running momentum is completely decoupled from the shove and gives them no advantage.

Page 173 from the PHB describes advantage and disadvantage in detail. In particular

The DM can also decide that circumstances influence a roll in one direction or the other and grant advantage or impose disadvantage as a result.

So, for example, if the DM finds that a Half-orc moving-and-shoving (similar to Charging) into a Gnome (which is small) should give the half-orc some advantage, ruling that the Athletics Check is made with advantage for the half-orc (or disadvantage for the gnome, since it is a contest) is not even a house-rule, it's just a ruling based on the book's RAW.

I.e., you don't need a written rule to grant advantage or disadvantage, the rules already cover the fact that the DM can grant them based on circumstances they find relevant.