Is the DAC the digital equivalent of a cartridge.

I mostly agree with this.  the importance of a DAC is underestimated by some.

The room and speakers are the instruments you hear from, but the DAC is how the signal gets created.

There are people who can hear through bad rooms and mediocre speaker selections but I sure can't.  While this is in a way a debate similar to "which would you rather have, a heart or a pair of lungs?" the room biases everything else.

It's a fun analogy, but I really don't know how true or significant it really is. Also, doesn't the gospel regarding analog components proclaim that it is the turntable...the thing that spins the record around...that is the most important?