GH-97950: Use new-style index directive ('builtin') (#104164)

* Uncomment builtin removal in pairindextypes

* Use new-style index directive ('builtin') - C API

* Use new-style index directive ('builtin') - Extending

* Use new-style index directive ('builtin') - Library

* Use new-style index directive ('builtin') - Reference

* Use new-style index directive ('builtin') - Tutorial