In Excel you have goal seek for this functionality. I believe it does some form of numerical solving of the equation system.
Good for every situation when you need to solve equations!
In the context of using spreadsheets I think about solving simple financial or maybe construction/mechanical design problems where you don’t want to solve it manually or program it and a spreadsheet is a quick and useful interface.
It does not. It perturbates the variables and uses a hill-climbing algorithm.
This is very different in practice, because it is pervasive rather than something you have to set up for particular cases.
If this was usual it would help a lot with people's tendency to hard code the correct answer rather than fix formulae. Just that aspect of it would be a huge improvement. People do this all the time with simple financial problem, for example.
A lot of what people use spreadsheets for is not all that simple. Again, especially with financial applications. People manage huge amounts of money using horribly complex models implemented in Excel.