(Also, http://www.theretheyretheir.com/. Not usually a grammar Nazi, but that's pretty hard to read)
Sure, solving renewable energy is a really valuable thing to do, but it's still only solving 1/3 of the problem.
Well, here in the Netherlands, the majority of the trains will soon (2018) be powered by 100% wind energy. 50% since this year.
So it's categorically (and I would think obviously) incorrect to say "you can't use electricity for transport".
Heating is another question entirely. You have more of a point there -- it's hard to find an alternative to natural gas for heating.
It's actually a comparatively easy problem to solve. Have a large boiler, heat that with electricity while you have an abundance of energy and keep it moderately well insulated. Drawing the heat off that is simple.
Storing electricity for usage as electricity is the primary problem.
Ivanpah has the capability of producing about 400 MW. It produces no energy at night, and doesn't max out the rest of the time. It rests on 4,000 acres of land. 
The Palo Verde nuclear plant can produce 3,939 MW from three reactors (currently operating at 3,875 MW). It's located on 4,000 acres of land. It can actually generate 3,875 MW, and can do it all day long if necessary. Most nuclear plants don't run at max capacity often, but it's irrelevant given the extreme difference, reduce it by 75% and it stomps Ivanpah. 
Then throw in the cost of covering, say, a square mile of land with solar panels versus the cost of building and operating a nuclear plant, even the ridiculous fantasy numbers of the OP. Solar looks pretty good.
The Kashiwazaki-Kariwa monster in Japan, with seven reactors, was capable of producing nearly 8,000 MW on just ~1,000 acres. Or upwards of 50 to 80 times what the best utility solar installations today can produce at max output on the same amount of land.
The Solar Star installation could end up being the best solar comparison right now (completed in June). At max capacity, it might narrow the ratio such that the Japanese plant is capable of producing 50 to 60 times the power on the same land.
579mwh ~= .6gwh, on 3200 acres. Assuming a larger 4000 acre site and some efficiency gains on this immature tech, a gigawatt seems reachable.
If a solar plant could hit 50% of the energy efficiency on the same land (and the poor-performing Ivanpah plant reveals that it's possible), that raises a question of why we need the cost and complexity of nuclear.
btw. i'm against nuclear and non renewable energy, however I think its hard to actually replace ever "bad" energy in the next 10-20 years. we will need a longer time for that. especially since building renewable energy plants fast isn't good for our environment either.
The highest capacity US nuclear power plant can out-produce the best utility scale solar installation by 10 fold, with both at max capacity, on an equal amount of land.
nothing to do with today. just a "what would happen on a perfect day". peak vs peak.
also peak of a nuclear power plant could be raised by human and solar panel peak is barely measurable, since there are a hugh amount of factors that could change it.
Solar is also mature by your lame definition of just being around for a long time. In fact, solar has been around longer than nuclear in that regard so it must be more mature then. /s