Refetch Sync dialog on focus and add spinner

[nightnei]

Related issues

• Fixes STU-1507

How AI was used in this PR

AI assisted me, but the implementation is mine.

Proposed Changes

1. The use case is - user opens Sync dialog, realizes that they need to connect Jetpack in Pressable, so they keep the dialog open and move to the browser to connect Jetpack to their site on Pressable. Then they return to Studio, don't see the just-connected site, and they restart Studio.

The solution - we need to refetch sites on the focus event, if the Sync dialog is opened in Studio.

2. Right now, when we reopen the Sync dialog - we take data from cache and start refetching, but we don't communicate that we are refetching them. That's why when a user reopens the Sync dialog, we have a flash of content after a few seconds. I had a case where I was selecting a site and ended up choosing the wrong one because of a flash of content.

The solution - we should communicate in UI that we are refetching sites.

Testing Instructions

1. Opean Sync dialog and assert that you see Loading sites
   
2. Close the dialog and open again
3. Assert that you see sites rendered from cache, but covered with blurred "Refreshing sites" spinner
   
4. Leave Studio and focus on another app on your laptop
5. Comeback and focus on Studio again
6. Assert that you see "Refreshing sites" spinner
   

[wpmobilebot]

📊 Performance Test Results

Comparing e1b30fb vs trunk

app-size

Metric	trunk	e1b30fb	Diff	Change
App Size (Mac)	1271.61 MB	1271.61 MB	+0.00 MB	⚪ 0.0%

site-editor

Metric	trunk	e1b30fb	Diff	Change
load	1894 ms	1881 ms	13 ms	⚪ 0.0%

site-startup

Metric	trunk	e1b30fb	Diff	Change
siteCreation	8181 ms	8191 ms	+10 ms	⚪ 0.0%
siteStartup	4861 ms	4847 ms	14 ms	⚪ 0.0%

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Results are median values from multiple test runs.

Legend: 🟢 Improvement (faster) | 🔴 Regression (slower) | ⚪ No change (<50ms diff)