Hdmi is the full pure digital output. From a monitor with a headphone jack it's the same exact signal as the hdmi would output except the jack is there as a convenience. Think 0 lag and 0 overhead from the video's signal. They are doing this for your convenience. "crappy monitor dac" simply does not exist. Even if your monitor had speakers built-in, the jack would bypass that completely. IOW, there is not a monitor made that has a 'dac' built-in. Why? HDMI is why. ALL the audio is just a part of the video stream/s file itself. IF the video and the audio were separate files, THEN you'd start thinking dac. This is why no one makes a headphone that uses an spdif plug. They don't need to.
This...Ok.
I know where you're getting at. But a DAC is a DAC is a DAC. The one thing that always bugged me about the "audio community", and I suspect it's causing confusion here, is the haze that it put around DACs.
If a monitor outputs an audio signal it receives from HDMI, it *has* to have an onboard DAC. Some onboard circuitry that converts the digital signal coming over HDMI to the analog signal speakers/headphones use.
Now, the second part of your post. Video and audio streams are always separated during the decoding process. The video stream is sent to the video output - monitor, TV, occular implants put in by little gray men in the Nevada desert, whatever. Audio signal goes to whatever is decoding tbe audio signal - DAC onboard the motherboard, DAC onboard the monitor, USB DAC, it's all "an output". What I'm getting at is, the original audio "source" is transparent to the DAC. It doesn't care if the source is an MKV file, an MP3 file, or a Youtube stream.
So while there are different transmission methods to the DAC - traces to a chipset on the mobo, USB to a standalone unit, HDMI to a monitor, TOSLINK to a receiver, etc - the end result is the same. A digital signal is received, a DAC - which literally means Digital to Analog Converter - converts it, and then outputs an analog signal.
To top this off, no, not all DACs are created equal. This is as obvious as reading spec sheets. Some can convert 44.1KHz files, some can convert 384KHz files. Some top out at 24bit, others "claim" to do 32bit. Some will offer cleaner sound than others. The DACs stuck in a monitor? Chances are, unless you have a super ultra high end monitor, is most likely an afterthought, and filtering is usually not that great. Even though you're getting the full uncompressed audio stream over HDMI, you can go plug in a $400 headset and still be underwhelmed.