The DAC inside your laptop is likely just fine but the headphone amplifier in the laptop is probably lackluster.
I bought one of these to use with my AKG 720 headphones which the built-in headphone amp in my laptop couldn't handle. Measures basically perfectly and has no trouble with the AKG headphones:
https://www.audiosciencereview.com/...cally-jm20-max-headphone-dongle-review.62260/
As far as "does the DAC chip drive the headphone out", do you mean is the DAC used to process audio on the headphone out? A DAC is the chip that converts digital audio signals to analog waveforms. It's not an amplifier. In the Toppings you mention (and in any other DAC/amp combo I'm aware of), the incoming signal is converted by the DAC to an analog waveform, and is then sent un-amplified to the pre-outs, and is also routed internally to the headphone amplifier to be amplified.