[RPG] Does “Disarm” basically allow you to steal

Probably not. Tavern Brawler requires you to "hit the creature" with a weapon attack. While the Disarm rule involves making an attack, it's not clear that that attack involves hitting a creature. Normally, hitting a creature requires beating its AC:

When you make an attack, your attack roll determines whether the attack hits or misses. To make an attack roll, roll a d20 and add the appropriate modifiers. If the total of the roll plus modifiers equals or exceeds the target’s Armor Class (AC), the attack hits. The AC of a character is determined at character creation, whereas the AC of a monster is in its stat block.

Since the attack you make to Disarm isn't against a creature's AC, it probably doesn't involve hitting a creature, and therefore can't trigger Tavern Brawler.

Yes.

As you noticed in the DMG, a disarm attempt is a hybrid attack/contest: attacker's attack roll against defender's athletics or acrobatics check.

As to your specific example, you've (basically) got it all right there, too. Let's step through it.

a lvl 1 Human Fighter with a shield and no other weapon could attack the defender with a shield (Tavern Brawler)

Yes. It'll be as an improvised weapon, but nicely for a tavern brawler he'll be proficient.

declare they are attempting a disarm before damage is rolled (atk vs skill)

I believe you would have to declare that the attack is a disarm attempt before the attack roll, not after the attack roll but before any damage, as this seems to imply. The reasoning: if you could see the attack roll, you know it doesn't beat the AC, then declare "oh, I was trying to disarm," then that stinks mightily of cheese.

and if successful disarm the defender

Yup, per "disarm" in the DMG,

then grapple as a bonus action

Yup, per tavern brawler,

and move the defender from their weapon by 15ft?

Yup, per "Grappling," PHB p.195 (assuming your base speed's 30').