Migrating to `System.Text.Json`

[just-ero]

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.

[yufeih]

Thank you @just-ero for offering help! This is definitely something we are willing to accept as a contribution. We appreciate your support.