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').