Forms: fix integrations modal flash

[edanzer]

Proposed changes:

On the forms dashboard, we recently removed the Integration tab and added an Integrations button that opens a modal. I noticed there's a flash when closing the modal. See video below. I think this is related to making the modal dependent on react routing. This PR fixes it by making the modal open/close state dependent on component state, while also keeping the /integration route so we can link to the modal.

The flash:
https://github.com/user-attachments/assets/59a2943f-eb6b-4636-8d00-f25dca3cb780

Other information:

• Have you written new tests for your changes, if applicable?
• Have you checked the E2E test CI results, and verified that your changes do not break them?
• Have you tested your changes on WordPress.com, if applicable (if so, you'll see a generated comment below with a script to run)?

Jetpack product discussion

None.

Does this pull request change what data or activity we track or use?

No.

Testing instructions:

Go to Jetpack > Forms. Click the integrations button in the header to open the modal. Then close the modal and confirm it is smooth with no secondary flash. Try it a few times to confirm, as the original flash often varied in severity.

[github-actions]

Are you an Automattician? Please test your changes on all WordPress.com environments to help mitigate accidental explosions.

• To test on WoA, go to the Plugins menu on a WoA dev site. Click on the "Upload" button and follow the upgrade flow to be able to upload, install, and activate the Jetpack Beta plugin. Once the plugin is active, go to Jetpack > Jetpack Beta, select your plugin (Jetpack), and enable the fix/integrations-modal-flash branch.
• To test on Simple, run the following command on your sandbox:

bin/jetpack-downloader test jetpack fix/integrations-modal-flash

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• In your local development environment, use the jetpack rsync command to sync your changes to a WoA dev blog.
• Read more about our development workflow here: PCYsg-eg0-p2
• Figure out when your changes will be shipped to customers here: PCYsg-eg5-p2

[github-actions]

Thank you for your PR!

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Follow this PR Review Process:

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Jetpack plugin:

The Jetpack plugin has different release cadences depending on the platform:

• WordPress.com Simple releases happen as soon as you deploy your changes after merging this PR (PCYsg-Jjm-p2).
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  • Scheduled release: December 2, 2025

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[CGastrell]

LGTM!

[jp-launch-control]

Code Coverage Summary

Coverage changed in 1 file.

File	Coverage	Δ%	Δ Uncovered
projects/packages/forms/src/dashboard/integrations/index.tsx	0/14 (0.00%)	0.00%	4 💔

Full summary · PHP report · JS report

_{If appropriate, add one of these labels to override the failing coverage check: Covered by non-unit tests Use to ignore the Code coverage requirement check when E2Es or other non-unit tests cover the code Coverage tests to be added later Use to ignore the Code coverage requirement check when tests will be added in a follow-up PR I don't care about code coverage for this PR Use this label to ignore the check for insufficient code coveage.}