Make syncing post types as part of syncing blog data

[crazytonyli]

Description

Do not show the "More" row if there are no custom post types on the site.

This PR also makes "Custom Post Types" an experimental feature.

[dangermattic]

	2 Warnings
⚠️	View files have been modified, but no screenshot or video is included in the pull request. Consider adding some for clarity.
⚠️	This PR is assigned to the milestone 26.7. The due date for this milestone has already passed. Please assign it to a milestone with a later deadline or check whether the release for this milestone has already been finished.

Generated by 🚫 Danger

[wpmobilebot]

📲 You can test the changes from this Pull Request in WordPress by scanning the QR code below to install the corresponding build.

	App Name	WordPress
	Configuration	Release-Alpha
	Build Number	30985
	Version	PR #25267
	Bundle ID	org.wordpress.alpha
	Commit	f774f5a
	Installation URL	2gbbgrrfnlqqo

Automatticians: You can use our internal self-serve MC tool to give yourself access to those builds if needed.

[wpmobilebot]

📲 You can test the changes from this Pull Request in Jetpack by scanning the QR code below to install the corresponding build.

	App Name	Jetpack
	Configuration	Release-Alpha
	Build Number	30985
	Version	PR #25267
	Bundle ID	com.jetpack.alpha
	Commit	f774f5a
	Installation URL	13glg63hduv2o

Automatticians: You can use our internal self-serve MC tool to give yourself access to those builds if needed.

[wpmobilebot]

🤖 Build Failure Analysis

This build has failures. Claude has analyzed them - check the build annotations for details.

[sonarqubecloud]

Quality Gate passed

Issues
0 New issues
0 Accepted issues

Measures
0 Security Hotspots
0.0% Coverage on New Code
0.0% Duplication on New Code

See analysis details on SonarQube Cloud