I agree with you to an extent. Whether the answers here support so called god (little g) methods or not I personally feel, just as you do, that methods should be precise in what they do.
That said, local functions can in fact support SOLID. One example is the ability to easily wire up call backs from events giving the method single responsibility, close the scope, and minimize error. This can be done with local delegates but it isn't as clean or straight forward.
Recursive local functions are also in line with SOLID. There are many situations where you have to write two methods to complete one operation because it's a recursive method and now you can put that logic all in one place. There are even times, depending on the data and types being passed around, that local functions will actually perform better due to the fact they are borrowing scoped variables.
This is only two quick examples that probably don't require a debate or lots of code to illustrate how local functions can be better and inline with SOLID but there are more.
Local functions can also easily polute your code and make methods hard to manage / read and tear away from SOLID principles. Just like any other C# component we have a responsibility to pick and choose when and why we use local functions and SOLID as well as other development practices / guidelines should be considered when we do. I don't agree with making methods longer, more complex, or having more than one job to do just for the sake of scoping methods to it.