During the playtest, there was a Charge action that all characters could use. What it allowed was that as an action, you could move up to half your speed, and attack any adjacent creature when you finished moving. This effectivley allowed you to have a movement of 1.5 times your normal movement rate, in exchange for losing extra attack etc.
Besides trying to figure out what half of 25" is, and being open to the odd debate about if it counted as an attack action for extra attack or not, it didn't negatively impact our game much. We did not add any bonus to hit, or a penalty to AC, but if worrying about those modifiers is not a concern, or if you enjoy that sort of thing, I think a +1 in either direction would be fine.
Compared to the feat, there is still a reason to take the feat since you gain an extra 10-15 feet of movement, and can also shove and not only attack. However, I would bump up the feat to also give a +1 to str to compensate for the new houserule.
Why would you Shove without a follow-up?
Knocking someone down is not really a good strategy in any fight (real or imagined) if you don't have a follow-up. When was the last time you saw a UFC/MMA fighter push someone to the ground and then just let them get up? When somebody in a gang does a shove, they're doing so to let their friends jump on the fallen.
Once you knock someone down, your next step is really to Grapple them or attack them while they are on the ground. A Grappled creature's movement is reduced to zero, so you can effectively prevent them from getting up (like the gang above)
In fact, the Monk actually do this Shove and Attack starting at level one with their Martial Arts skill. So you knock somebody down (Attack Action), kick them in the head (Bonus Action) and even step away if you want (Move Action, their AoO at Disadvantage). If you move away, they typically can't attack you on the following round because they have to waste their movement standing up.
Note that at 5th level a Monk (or other PC with Extra Attack) could do both a Shove and Grapple as part of the Attack Action to effectively pin their opponent. Escaping the Grapple requires an action and standing requires half movement, so pulling this off pretty much wrecks the target's turn.
This is clearly a gang-up tactic, but that's expected.
Is Initiative the Problem?
Consider that the problem here may actually be the initiative order itself.
You point out the problem of effectively "metagaming" the timing of the action, but also the problem that it will basically never allows you to benefit.
DMG (P.270-1) presents an alternate initiative system titled Speed Factor.
Under this variant, the participants in a battle roll initiative each round. Before rolling each character must choose an action...
This is followed by some modifiers for speed and action type. The action order in each round is random (weighted). This means that knocking someone prone will always have at least a chance of being useful. And there's no "house-ruling" required as this is straight from the book.
If you're really worried about the metagaming aspect of Shoving/Grappling etc. this might help as well. I've used this "initiative per round" and it does kind of solves the problem of gaming the initiative order, both for things like Shoving and for things like Healing. However, it introduces other problems like wasted Actions. So it may be a little much just to solve the Shoving problem.
Best Answer
First, I want to point out that flying characters already had this option at the cost of a single feat, Flyby Attack. Flyby Attack is different from Spring Attack in that it allows any standard action during movement, which is massively more valuable.
So this would eliminate the point of a few feats, like Flyby Attack and Spring Attack. But I have played in games where Spring Attack was explicitly changed to work like Flyby Attack (rather than being a single attack only) and had its Dodge and Mobility requirements waived, and it still didn’t get taken by anyone. I have also played characters that did take it, because it was required for something or other, and... it didn’t really matter.
In the end, positioning just isn’t that important in the d20 System (unlike, say, 4e). Ranged characters don’t have particularly stringent demands on where they stand, and melee characters typically don’t want to leave melee once they’ve closed.
So the only cases where it ends up mattering a lot are:
your melee attack kills, so you want to move on to the next thing. Ultimately, to have your one single attack be the difference isn’t that common an occurrence, and you can always just charge next turn or whatever. This can actually hurt you if you move into melee with your next target, but cannot attack – now that target gets to attack you first, without even having to move. Bad news bears.
making the difference in an otherwise-symmetrical situation (move in, attack, move out). Unless you foresee a lot of literal mirror-match one-on-one duels on featureless terrain, usually some other factor is more significant.
you have truly massive movespeed and the room to use it, and you have powerful ranged options (read: spells). By which I mean more than double that of any of your opponents. That’s pretty hard to get, and usually costs too much. But it does make a big difference for, say, dragons..
Personally, I say go for it. It offers a few more options, it seems a little more sensible, and the loss of these feats as meaningful choices is minor; they probably didn’t deserve to be feats in the first place. Just make sure to change other feats’ requirements accordingly.