New Horizons, to use your example, weighed a thousand pounds and used a 2 meter dish transmitting at something like 12 watts to compensate for the fact that the receivers are billions of miles from earth and hidden beneath a blanket of RF noise. The inverse square law can't really be beaten at that kind of distance so everything becomes inefficient by design.
If we can pick up that tiny 12 watt whisper of a signal from billions of miles away, surely we we could design much lower power omnidirectional signals that relay between mesh nodes closer together using far less power?
Imagine a string of probes that are all within a few thousand miles of each other with clear line of sight. Yes, we might need six million of them to cover that same distance, but if they were cell phone sized devices produced using what we've learned about consumer electronics it should be feasible to just keep launching them forever, for a few hundred bucks apiece, until we eventually build a large network that could assemble high resolution data by combining multiple sources.
We keep trying to fight the rocket equation, but that's not a battle that can be won. Mass is always going to be the limiting factor for space exploration, so maybe we can just start launching lots of intelligent low mass things regularly instead of the occasional big dumb thousand pound lump of metal.
Each one needs whatever sensors, but more importantly, the auxiliary stuff to last years or decades in transit: in particular power supplies, heaters (=power!) if you can't make your electronics survive constant cryogenic temperatures, as well as comms amongst themselves to organise the mesh and high-gain comms back to Earth.
Maybe the worst of that could be solved with a nuclear power supply and then it's basically "just" radio and software design.
I also don't think you'd use onmidirectional mesh comms, you can get a lot of milage (literally) out of a phased array that can steer the beam at each target, plus it also becomes a bonus multistatic radar network.
I suppose that's my point. It's really just a series of maybe-not-that-easily solvable engineering problems, and it would allow us to not only explore further but to do it at a relatively low cost and with the ability to "upgrade" the network gradually as each generation of probe improves. More importantly, it would allow us to finally get around that pesky rocket equation and do it cheaply enough that we might actually get political buy-in.
> nuclear power supply
This was my exact thought, a small RTG power supply in each could provide enough power for hundreds of years with no moving parts. Not enough for billion mile transmitters anymore, but now they don't have to be.
> phased array that can steer the beam at each target
That's a great idea, and like most of the individual pieces of the plan it's kind of a thing we already know how to do. Sure, you could dedicate the next decade to solving it well, but you COULD solve it today with variations on off the shelf systems.
Really, the power source and antenna design are just a few of several hundred (thousands?) of engineering problems that would need to be solved, but all of the engineering challenges I can think of are solvable with variations on current tech.
The only reason it isn't being done is that nobody is doing it.