1. Push 5 feet is a
much narrower effect of Shove compared to knock prone. It's situationally useful, has rider effects defined by the DM, and will rarely provide more benefit than "advantage on all melee attacks until start of next turn and half movement speed until end of next turn" without a situational ruling from the DM.
2. I specified the Dual Wielder feat, which explicitly removes the Light weapon requirement. Beyond that, your argument here is essentially that it doesn't work
at level 4 or for variant humans before level 5. It's a pretty rotten feat if it's only better at level 4.
2b. This is the scenario I'm presenting. Using a RAW reading, Shield Master is required to do Attack + Shove and Dual Wielder/Two Weapon Fighting is required to do Shove + Attack. However, given that Shove + Attack is better in a vacuum because the shove can benefit the character doing it, Dual Wielder is the better option for a shoving character.
Of course, if you're going for RAW readings, you get to entertain the argument that
a shield is an improvised weapon and, therefore, Dual Wielder allows you to actually
attack with a shield, making Dual Wielder the better shield user overall. Sure, you need Tavern Brawler to get proficiency to hit, but the additional +1 to AC on top of the sheild's +2 is really pretty ridiculous even in the face of evasion.
Sure,
Crawford says that wasn't the
intent and some later tweets
avoid answering at all, but when Crawford uses the "I" word, you pretty much know that the RAW support the more absurd reading. That is, of course, why a flat strict RAW reading is a bad idea in general.