Have you considered electroless copper plating? This is catalyzed by an activated surface, or autocatalyzed by a metal surface (so it continues to build up while it soaks in solution). Any conductive ink could be used, or ink need not be used at all if an appropriate activation bath can be found. (Which, I think they do this with FR4 commercially, anyway, so this should be practical?)
I don't know how practical such a process is for a printer; you'd probably need a reservoir of starting reagents, a bath where they mix with the board (handling should be pretty easy, just teflon-lined grippers and stuff, if nothing else?), then a waste disposal tank where the spent materials can be stored for disposal (or processed for recovery, or for easier disposal e.g. evaporating solvent to concentrate it for reuse, ion exchange to remove/recover heavy metals (copper), evaporating to solid for dry disposal, etc.).
Hm, I suppose all those things are what the industrial proto printers do already, wouldn't it be? A few extra steps (tin resist, electroplate, etch, strip..), but those could be omitted for a somewhat lower quality process. In which case you'd still have basically the same thing, and consequently it wouldn't really be cheaper, huh? So I guess your intent is to avoid chemical processes if at all possible?
I wonder if PVD something or other would ever be competitive. Drawing a hard vacuum is a pretty high bar to start off with (the hardware alone would probably account for 1/3 your target cost..), but if you can manage that, repeatably and cheaply enough, it's not nearly as messy, I don't think? You'd feed in, either copper plate for sputtering, or wire (to a tungsten/molybdenum boat) for evaporation. Still need a resist and/or etch step though, hmm.
I wonder if a plasma process could be employed for the etching, too? It wouldn't be practical on thick layers though, so you'd have to start very thin (~um) regardless. Which isn't very helpful compared to, say, a conductive ink, which does the same basic thing with a lot less bother!
And that's to say nothing of multilayer, which none of this is any help with.
So that sounds like, working on inks is the right direction, which is encouraging.

Flash sintering sounds awfully cool (hot); how do you control the temp rise in the insulator/substrate and conductor? Would that be a matter of picking exposure wavelengths versus absorbance spectrum? Doping with susceptors (e.g. graphite) or reflectors (e.g. TiO2 white pigment)? Alternately, spreading out the power density through depth, by using a much more translucent insulator than the conductor?
If it's not so easy to control, would it be necessary to raise the insulator melting point instead? Flash sintered ceramic or glass would be awfully cool. Harder to handle (brittle), but another interesting product nonetheless -- consider the cost of existing ceramic hybrids. Smaller market I would suppose, but still a problem in need of a solution? Would also allow better consolidation (final step in a kiln; should be a simple enough physical process, at worst needing an inert or reducing gas fill, no liquid or solid chemicals and no additional waste?).
Huh, does flash sintering not have any problems with the substrate curling up like a potato chip?
Anyway, just thinking out loud. This sounds really cool, and I wish you the best of luck on it!

Tim