Migrating to `System.Text.Json`

## Description

As mentioned by @yufeih in https://github.com/dotnet/docfx/discussions/9717#discussioncomment-8563299, the use of `NewtonSoft.Json` in this library is due to the age of it. That's understandable.

However, using .NET's own `System.Text.Json` namespace should provide at least these obvious benefits:
- Clear performance advantage.
- One less third-party dependency.

I propose that a gradual move to `System.Text.Json` be made, despite it not being trivial. This move can coincide with the change to providing public config/options classes proposed in #9726.

Feel free to close if not planned or move to a "Backlog"-type milestone.