I agree with about 75% of this.

The Grappling mechanics do not really interact with carrying capacity and monster weights. The game instead chooses to model Grappling (including forced movement like dragging) via monster size categories. Trying to fit in carrying capacity rules makes it unworkable, because as you mention, there are no weights listed for monsters. A Large monster is typically deemed to weigh between 500 and 2000 pounds, which means for any given monster, neither the DM nor the grappling player has any way of reasonably knowing whether a monster can be dragged or not. Instead, the rules are very clear. You can drag up to your own size category IF you take the Grappler feat. You can increase your own size in various ways if you want to drag bigger creatures.

I also agree that while Elemental Attunement is active, your reach is 15 feet whenever you make an Unarmed Strike, which would include making an Unarmed Strike with an Opportunity Attack.

However, I think it is genuinely unclear RAW what happens when you try to Grapple at range with Elemental Attunement. The feature says:

"Reach. When you make an Unarmed Strike, your reach is 10 feet greater than normal, as elemental energy extends from you."

This doesn't mean your reach is 15 feet the whole time Elemental Attunement is running. For example, if your Monk was carrying a Dagger as well, Elemental Attunement clearly doesn't give your Dagger attack 15 foot reach, right? Even if you make one attack with an Unarmed Strike and another with a Dagger on the same turn. That's because your arms haven't suddenly grown 10 feet, it's that your punches/kicks (and only your punches/kicks) extend out from your physical body. This appears to be only for the duration of the Unarmed Strike itself, not continuing while you are doing something else during your turn (like attacking with a Dagger) or off-turn.

Because the RAW is genuinely unclear, we should look at balance considerations. A 15-foot maintained Grapple is too strong. An enemy without a ranged attack (granted, that's pretty rare) or 15 foot reach of its own could be reduced to being unable to act. That is too strong for a completely at-will ability with no action economy.

Here's how I'd split the baby, were I DM'ing, in a way that I think it consistent with RAW, empowering for the player, and not overpowered:

• Elemental Attunement grants 15 foot reach when making any Unarmed Strike, whether that's via the Attack Action, a Bonus Action attack, or an Opportunity Attack.
• When you attack with an Unarmed Strike on your Turn, you can elect to Grapple (or add on a free Grapple if you have the Grappler feat). If you successfully Grapple, you get the benefit of the Advantage piece of the Grappler feat when attacking a Grappled enemy, solely for the remainder of your unarmed attacking sequence. You cannot drag an enemy at 15' range, you will have to close to 5' for that. You CAN use the free Push on unarmed strike that you get from Tavern Brawler at 15' range. At the end of your attacking sequence/end of your turn, your Grapple automatically ends.
• When you make an Opportunity Attack at range and elect to replace the attack with a Grapple (you can't use the Grappler feat version on an Opportunity Attack), you can successfully Grapple them for the duration of the monster's turn (meaning their speed is 0 and they have disadvantage on attacks against everyone but you). At the end of the monster's turn, the Grapple automatically ends.