I'm trying to find the mechanics for a running tackle / charging attack, but can't seem to find one in the PHB or the DMG.

I'm looking for something that allows players to run towards a creature and use their momentum to knock it prone (or push it back) and maybe even deal a little bit of damage.

Simply put: How can a player character clothesline another creature?

Charger

I know that there is a Feat called Charger:

Charger: When you use your action to Dash, you can use a bonus action to make one melee weapon attack or to shove a creature. If you move at least 10 feet in a straight line immediately before taking this bonus action, you either gain a +5 bonus to the attack’s damage roll (if you chose to make a melee attack and hit) or push the target up to 10 feet away from you (if you chose to shove and you succeed).

However, Charger requires a Feat, making it pretty hard to obtain.

Move + Shove

I also know that players can just use their regular movement coupled with the Shove action. But that seems rather anticlimactic because the Movement and Shove action are completely independent from each other. In other words, the player's running momentum is completely decoupled from the shove and gives them no advantage.

Am I missing something obvious from the rules or does this mechanic just not exist?

Thanks!

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 - Feracious Charger

Additional to the charger feat, the Cavalier subtype 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.