Add Reconnect button to embedded disconnect overlay and restore VM service connection after sleep/wake

[rishika0212]

Fixes #9683

This PR adds a Reconnect button to the embedded DevTools disconnect overlay and fixes reconnect behavior after a machine sleep/wake cycle.
Previously, when the system went to sleep during a Flutter debug session, DevTools lost its connection to the VM service and displayed:

"Disconnected — Run a new debug session to reconnect."

In embedded/IDE mode this message provided no actionable way to reconnect, forcing users to restart their entire debug session.
Standalone DevTools already included a Reconnect button, but the reconnect flow only attempted to restore the DTD connection and did not re-establish the VM service connection, causing reconnect attempts to silently fail.

This PR improves the reconnect behavior by:

• Capturing the last VM service URI before disconnect
• Adding a Reconnect button to the embedded disconnect overlay
• Displaying a loading spinner during reconnect attempts
• Updating the reconnect logic to call FrameworkCore.initVmService() directly instead of routing through routerDelegate.navigate()

Using routerDelegate.navigate() previously routed through _replaceStack → manuallyDisconnect().
If reconnect failed, manuallyDisconnect() set userInitiatedConnectionState = true, which caused the disconnect overlay to be permanently suppressed. Users were then unable to retry reconnecting without restarting the debug session.
With this change, reconnect attempts directly restore the VM service connection, allowing DevTools to recover without restarting the debug session.

Testing
Automated tests
Updated disconnect_observer_test.dart:

• Expect a Reconnect button in both embedded and standalone modes
• Updated overlay text to match the new UI copy: "Or run a new debug session to reconnect."

Manual testing
The following scenario was tested:

• Start a Flutter debug session
• Open DevTools Inspector
• Put the system to sleep
• Wake the system
• DevTools shows Disconnected
• Click Reconnect

Result:
DevTools reconnects to the VM service
The debug session continues without needing to restart the app

Release Notes
A note has been added to:
packages/devtools_app/release_notes/NEXT_RELEASE_NOTES.md

Pre-launch Checklist:

• I read the [Contributor Guide] and followed the process outlined there for submitting PRs.
• I read the [Tree Hygiene] wiki page, which explains my responsibilities.
• I read the [Flutter Style Guide] recently, and have followed its advice.
• I signed the [CLA].
• I listed at least one issue that this PR fixes in the description above.
• I updated/added relevant documentation (doc comments with ///).
• I added new tests to check the change I am making, or there is a reason for not adding tests..

[elliette]

Thank you for the PR! Please address the comments and I will take another look.

[rishika0212]

Pushed updates addressing all the comments. Thanks for the helpful feedback.

[elliette]

A few more changes but LGTM! Thank you

[elliette]

One small change but otherwise LGTM!

[auto-submit]

autosubmit label was removed for flutter/devtools/9693, because This PR has not met approval requirements for merging. The PR author is not a member of flutter-hackers and needs 1 more review(s) in order to merge this PR.

• Merge guidelines: A PR needs at least one approved review if the author is already part of flutter-hackers or two member reviews if the author is not a member of flutter-hackers before re-applying the autosubmit label. Reviewers: If you left a comment approving, please use the "approve" review action instead.

[srawlins]

Love this, thanks!

[elliette]

Please run dt fix-goldens --run-id=24064811951 and upload the golden changes introduced by this change. This should help us determine whether the changes are expected or regressions. Thanks!