One part of the problem is NOT going to go away however - they have to pay to maintain the lines. Right now, that cost if covered by your electric bills. As the amount of electricity you draw from their generators goes down, they're going to reach the point of needing to charge you a flat fee just for the connection to the power lines, plus the usual fees for actually using their electricity.
If government did it, it would be extremely inefficient like all government services. We would all pay for power lines even if we don't use them via taxes (unfair), plus it would be run by political interests rather than economical ones.
Yes, it needs to be economical for someone to build and maintain a powerline, otherwise nobody would do it.
But this has an easy solution: Suppose you own the power lines, all you have to do is sell electricity at one price, and buy it back at a cheaper price call it the "bid" and "ask". The difference would be used to cover power line maintenance and new constructions. You would simply spend less in producing electricity and let users produce it for you. You are essentially getting paid to move electricity from one house to another.
What happens if too many people have solar and you have a surplus of electricity? you would lower the cost of the electricity. The cheaper your electricity is, the less people will install solar panels. You can keep the system balanced with no surpluses or shortages simply by adjusting the prices of electricity.