Unless I am mistaken, given two attacks, it is always better (for the fighter) to attack twice than (try to) shove a target prone and attack it once with advantage.
You are mistaken. (Sorry, couldn't resist after the starting line of "Unless I am mistaken...").
But the statement that it is
always better is wrong.
For just damage, two attacks are better than one with advantage (assuming no bonus that only applies to a single attack).
However, that looking only at damage the fighter deals. Let's unpack that.
First is "looking only at damage". There are times when denying an opponent half their speed is a huge benefit. An escaping foe with a faster move you are slowing down to stay within range. Heck, one of your speed that you're slowing down so the rest of the party can catch up. More speed allowing the foe to interact with hazards, or get to cover, or back among defenders or other safe place. Those are situational, but any situation where they come up breaks the "always" part of the assertion.
But more importantly, D&D is a team game. Assuming random distribution of initiatives (and a fighter can easily capitalize to make this favor the PCs even more), half your party will go before the proned foe. If there are more melee attackers, they will also have Advantage. Making the shove much more damaging to the opponent.
Standing up from prone consumes half of ones movement. Therefore, standing up is kind of moving, and movement does provoke attack of opportunity within the threatened area otherwise.
This is also wrong. Only movement OUT OF your threatened area provokes an attack of opportunity. A foe can dance around the 8 squares adjacent to a PC, moving freely without triggering attacks of opportunity.
If the prone target chooses the stand up, the fighter would kind of get his/her second attack back (with advantage!). Is this even overpowered? I am trying to make this option more worthwhile and more commonly used.
This is massively unbalanced.
Without shove: 2 attacks, no advantage.
With shove: 1 attack with advantage. Allies attacks with advantage. Reaction opportunity attack with advantage. Allies opportunity attack with advantage. Foe denied half their move. Even without allies it's going from two attacks without advantage to two attacks with advantage plus halving their movement. When you include even one ally who came to pound on the fallen foe they are getting a full attack an an opportunity attack, all with advantage.