Channel separation on a 3.5mm audio jack
In a 3.5mm headphone jack, the ground is shared between both channels, which means bridged amplifiers cannot be used. A bridged amp requires a separate connection for each driver, in stereo this means 4 wires.

With only a single LiIon cell as power supply, thus 3.0-4.2V. Minimum supply voltage being 3V, the headphone "ground" would be driven to mid-supply and both channels driven by an amp which would only be capable of +/- 1.5V relative to this "ground". This is loud enough for me, but maybe not for everyone.

An alternative is to use a DC-DC converter inside the phone to generate a negative supply (say -3V) and have each amp capable of outputting +/-3V for 2x the output swing and 4x the power.

Consider 300 ohms headphones. With 1.5V peak voltage, we get 7.5mW peak power. For example Sennheiser HD600 will produce 97dB at 1mW, so we get 106dB peak power.

32 ohms earbuds are a better match since the lower impedance allows more power with less voltage. So, with 1.5V on 32 ohms we get 70mW. With 90-99d/ at 1mW sensitivity this gives 108-117 dB peak.

With 106dB peak you can have 90-95dB average power which is already VERY loud at 1kHz. However sensitivity is measured at 1kHz which hides the fact that tiny earbuds are notoriously bad at low frequencies (ie, you get no bass).

So... that's not a clear-cut case. There is enough power even with a 3V power supply to make you deaf already.

So how could headphones not using 3.5mm jack sound better?


  the stereo sound was compromised due to poor sound channel separation 


That's bullshit. Nobody ever complained about super-expensive audiophile headphones having bad channel separation because of the jack. Problems with the jack are flimsiness and bad contacts, sure, but not channel separation. And let's not talk about channel separation on vinyl or tape, which is pretty bad, and never bothered anyone.


  and the sound quality was compromised due to a mismatch between phone and headphones.


That's the voltage issue, which isn't really an issue.

I can think about several reasons to remove the jack:


Sell new headphones. Profit! 
Have fanboys brag about their minuscule $200 bluetooth earpods (free publicity) and sell new ones when they get lost. Profit!
Remove audio circuitry from the phone. A good headphone driver adds to the cost. Profit!
Makes the phone thinner. Sure, you'll have to wrap it in a thicker protective case to prevent it from bending but... you still get the bragging rights!
Besides that... 


There is one reason I can think about that would make USB-C or bluetooth headphones sound better: the DAC is in the headphones instead of being in the smartphone. This means there can be a better DAC maybe, but more important, the manufacturer can add a DSP to do some frequency equalization. This means they don't have to make flat frequency response hardware and can use other kinds of engineering compromises between mechanical response and DSP correction. This could either be used to sell crap cheap hardware made tolerable by DSP, or to make the really good stuff.

An important feature that is made available is active noise cancellation in the phones without having to add an extra battery, since USB-C provides power.

From an engineering point of view, it makes sense to have the headphone hardware, loudspeakers, and the circuit designed to drive them and correct their frequency response packaged as a single unit.

From a sales point of view, you will buy new bluetooth phones every 3 years when the lithium batteries inside reach end of life, so that's awesome! Also it will make your USB-C connector break more often, which is great for profit too, since you get to buy a new phone.

Now you may say, the DSP bits could be implemented with an app in the phone... but where would the profit and vendor lock-in be then?

I'll keep my Sennheisers HD6XX though, thank you very much.