Electricity grids simply follow Ohms Law

Only if you do Ohm's Law in complex arithmetic, where Z = R + jωL - 1/(jωC).

one can put the grid underground for approx. ten times the cost of high voltage lines on towers.

We'd be better off going with superconductors.

The other part of the solution is simply to pay a few cents more per kWh for a more stable grid.

The grid can handle a lot more energy than it does; the problem is that it's sized for a peak which is reached for only part of the day.  Instead of spending more per kWh, we should spread our consumption around the clock and get more out of each dollar of equipment.  We might even be able to spend less, when all the savings are added up!

I am a physicist who does the job of a EE and has to deal with RF circuits all the time. Of course Ohm's law involves complex numbers for me. It can't be any other way. Ohm's law does not require the impedance to be real. It only requires the relationship between voltage and current to be linear. Which, by the way, it is not due to self heating of wires. But by the time you get to the point where that matters, your wires are usually damaged beyond repair. So I did not talk about it. :-)

Superconductors require power for cooling and lots of it. I have a whole book with a government report on the use of high Tc suprconductors for energy transmission and storage purposes. As far as I remember the technology of the early 1990s was nowhere near the engineering requirements for succesful superconducting transmission lines and generators etc.. I could be wrong but I don't think superconductors are quite there yet. In the future, maybe, for now... don't think so. Superconductors, IMHO, are just another one of those "magical" solutions to the public which are suposed to make the pain of having to pay for proper engineering solutions to go away.

As far as losses are concerned, the premier ways to minimize losses are to raise the voltage and generate locally. PV is a great way to cut I2R losses in summer when there is peak demand due to AC. We have close to 10% losses in the current system, most of which must be due to peak loads (that is simply plysics). If we can reduce peak loads, these losses will go down dramatically. That they exist in the first place is also good indicator that utilities have not kept their grids well maintaned. Which would be a political problem (due to deregulation, I suppose). It is not a technical problem, for sure.

Old houses, by the way can add 5% losses due to their wiring alone. Running loads on 120V is another one of those nonsensical US standards. It won't save your butt when the hair dryer falls into the bath tub but it will cost you four times the amount of copper compared to Europe's 240V standard to transmit the same power with the same losses.

"The grid can handle a lot more energy than it does; the problem is that it's sized for a peak which is reached for only part of the day."

True but currently local energy storage is by far less efficient than energy transmission. There is little to gain there with current technologies. Your assumption that the capital cost of the grid is a major cost driver for energy prices is also wrong. Operating the grid costs a couple of cents a kWh if its done right. If you doubled it in size, it would still be a lot less than the cost of environmentally friendly and sustainable energy sources. But I suppose that selling more energy while overloading the net beyond safe engineering limits is more lucrative to operators than helping customers to conserve and upgrade the net to be rock solid. Again, the solution to that requires social engineering, not EE. Copper, transformers, towers you can simply buy. These are catalog items. The will to invest, is not.

currently local energy storage is by far less efficient than energy transmission.

Not true in general.  Consider:

  • Ice storage A/C.  By operating at night when air temperatures are lower, the storage system can move more heat per kWh.
  • PHEV's.  These have no alternative to storage, so no efficiency comparison.

Peaks and slumps are both problematic for the grid; peaks stress equipment and shorten its life (as well as creating the conditions for cascading failures), while slumps leave the investment not paying what it's worth.  Levelling the load is good for both of those things, and a grid with a 30% margin day and night is better than a grid with a 5% margin during peak hours and 60% at 3 AM.