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.

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.

It does not. It perturbates the variables and uses a hill-climbing algorithm.