Could Bitcoin eat Counterparty's lunch?

How feasible would it be for the Bitcoin devs to duplicate Counterparty’s functionality without the XCP coin?

There are various approaches to do something similar and each has its advantages and disadvantages.
You can see the why’s and how’s in previous discussions (colored coins, sidechains, etc.) and a summary can be found here: What about sidechains, etc?

From what I understand, the only ability XCP has which could not be duplicated with BTC is its ability to escrow. How much of Counterparty’s functionality relies on the ability to escrow?

Some do, like its Decentralized Exchange and betting.
Counterparty uses bitcoin for all this, none of the transactions happen off-the-blockchain. Sure some implementations of colored coins may get similar features down the road, but Counterparty will get new features down the road as well.

There is no reason that there be just one implementation of certain Bitcoin 2.0 features. And you can fork Counterparty today and create your own implementation if you want.

I lost track of the decentralized exchange. Is it working currently?

You can access it in any Counterwallet.

Because most people are used to trading on a centralized exchange, a decentralized exchange - which requires several blocks to execute a trade - is considered less convenient and it often happened that users attempted to trade on it without understanding how it works (and the way it works is you’re supposed to stay logged in until the deal is settled, because Counterparty can’t take BTC from your address to pay - that part has to be executed by the person using BTC). So with users using “fire and forget”, trades would not settle and they considered that inconvenient.
That’s why last November BTC-related buy/sell was removed from Counterwallet interface to the DEx.
Having said that, you can still use the DEx to sell BTC from the CLI or API - the protocol still supports it. But because it’s semi-hidden, not many people do it.
The way it works now, you can buy and sell one Counterparty asset for another. That works fine, albeit the liquidity is low.