Fix CPT not immediately displayed in the "Publish" section

[crazytonyli]

Description

There are two sections that almost look exactly the same: Content and Publish. This PR brings in the same code in the Content section over to the Publish section.

The changes were included in #25300. This PR cherry-picks it to the releas branch.

[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

[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	31161
	Version	PR #25309
	Bundle ID	org.wordpress.alpha
	Commit	c570205
	Installation URL	7qnpdcqjlc1ng

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	31161
	Version	PR #25309
	Bundle ID	com.jetpack.alpha
	Commit	c570205
	Installation URL	2o042nr91e2cg

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.