TL;DR: power saving with a lid comes from the fact that you can use a lower power-mode on
your cookstove. Energy used to bring the water to the boiling point is roughly the same with or without the lid, energy saving occur after that point.
I'm afraid that the question and the current discussions are quite ambiguous on one capital point.
Just a matter of money; you can boil all the water you want with enough focusing solar collectors . There is an installation ( maybe more) in AZ (?) that melts metal with this method. Reliable,good ,safe, small ,etc are all a matter of judgement.