Wow, thank you! That does exactly what I was looking for and had previously thought was impossible.
For anyone else wondering what this means, the magic here is the Output Source that doesn't seem to have much documentation. It has a number of channels which I don't fully understand and will need to mess with more. But the net is that channel 0 of the Output Source without the DSK contains what's in the Program view (e.g. what you area stream). Probably for normal streaming, this would not be useful because since channel 0 is what you're streaming, it wouldn't make sense to then further generate any scene content which could, by its nature, not be streamed.
However, in conjunction with something like Source Record or the NDI Output filters, you could add additional overlays on top of the broadcast stream, and then send that somewhere else.
There is even more magic that can happen in conjunction with the DSK plugin. Channel 0 of the Output Source shows what is in the Program view
before the DSK is applied, and channels >= 7 have the DSKs (e.g. if you have 1 DSK, that is in channel 7, if you have 2 DSKs, the first is in channel 7, the second is in channel 8). That means using the DSK, you can do some things that you can't do by copy/pasting an "Overlay" source into every scene.
Using these capabilities, my use case of recording the current scene prior to applying the DSK is simple, and only just the beginning.
If you have dynamic content in the DSK (like chat browsers), you could record those overlays separately. If you had a live control room with multiple monitors, you could create a scene that contained the stream output with additional monitoring data overlaid (e.g. from browser sources that contained other live stats) and send that via a Projector to a monitor. Or you could through a normal source projector, send the chat overlay to one monitor, and then send a composite of the clean stream view + monitoring stats over to another monitor.
This is amazing!
@jkao you can add a source record filter to an output source on channel 0