I read that most modern VHF radios have DSC and since the liveaboard is a custom built boat that started operations in 2015, you would think they would have a fairly modern radio. My question is: Is it likely that they do have DSC and don't know it's there or not have it at all?
Correct. DSC has been required on all VHF radios sold in the US for quite some time (1999, I think). Not sure about other areas, though manufacturers may have phased out non-DSC radios in order to keep configurations down.
That said, despite almost all the radios out there having DSC capability, that doesn't mean they have enabled DSC. The vast majority of recreational boaters have not hooked up their DSC radios to take full advantage. Hopefully commercial boats have a higher take rate, but I'm not sure. To take full advantage of a DSC-enabled radio, you need to do two things. 1 is fairly easy, the other is a bit more difficult.
First, you need to get an MMSI and program it into your radio. This information would be sent to monitoring authorities in the event that the red button is pressed. Other information may or may not be sent depending on if the second thing is done.
Second, you need to connect the VHF to a GPS signal. Some VHFs have internal capability, so that part may be done. Even if this is the case, it's still a good idea to connect to a GPS chart plotter. Most chart plotters would be able to receive the distress signal and mark it on the screen provided the GPS is properly connected. Getting the GPS connected to the VHF is not the easiest thing in the world. Usually, this is done on the NMEA 0183 lines. Unfortunately, there hasn't been a standard color coding between VHF and GPS manufacturers for this. Basically, you need to connect the TX + wire on one device to the RX + wire on the other device, and vice-versa (sometimes, depending on GPS) The - versions of those wires also need to be dealt with. Basically, you need to get out the manual of both devices, and follow both sets of instructions. It can be a pain to do properly, so apart from my own boat, I don't have a lot of faith that it was done.