Fix: Multiple overwrites on rest_controller_class for wp_template/wp_template_part.

[jorgefilipecosta]

With the changes in #45215, we have two filters overwriting rest_controller_class of each other. This PR joins the changes in a single class and fixes the overwriting.

Testing

I created a new template and verified the add template modal appears. On trunk, it does not appear.

[github-actions]

Flaky tests detected in 037d509.
Some tests passed with failed attempts. The failures may not be related to this commit but are still reported for visibility. See the documentation for more information.

🔍 Workflow run URL: https://github.com/WordPress/gutenberg/actions/runs/4205049697
📝 Reported issues:

• [Flaky Test] should always expand single line selection #39787 in specs/editor/various/multi-block-selection.test.js
• [Flaky Test] should handle Enter across blocks #39763 in specs/editor/various/multi-block-selection.test.js
• [Flaky Test] should leave a scheduled page scheduled if canceled #47499 in specs/editor/various/switch-to-draft.test.js
• [Flaky Test] should make the modal content focusable when it is scrollable #48067 in /test/e2e/specs/editor/various/a11y.spec.js
• [Flaky Test] should revert a scheduled page to a draft if confirmed #47500 in specs/editor/various/switch-to-draft.test.js
• [Flaky Test] should write over selection #39938 in specs/editor/various/multi-block-selection.test.js

[TimothyBJacobs]

Code wise this seems good to me.

[jorgefilipecosta]

Code wise this seems good to me.

Thank you for reviewing 👍

[oandregal]

This PR improved on #45215 whose discussion to be backported is happening at https://core.trac.wordpress.org/ticket/57704#comment:4 It missed 6.2 and will probably miss 6.3. Follow trac ticket and original PR for discussion.