Site settings: Options don't sync after multiple changes in a row

### Impacted plugin

Jetpack

### Steps to Reproduce

1. Pick any Atomic site or Jetpack-connected site
2. While in Calypso, navigate to _Settings → Writing_
3. Change one of the settings, e.g. _Limit feed to excerpt only option_ (`rss_use_excerpt`)
4. Review the option in _Site Explorer_
5. The option should be in sync with what we set in the Calypso UI
6. Repeat the steps 3 and 4 until the option gets out of sync

Related discussion: p1667228637435869-slack-C011BCGRMBR

### A clear and concise description of what you expected to happen.

The site option user saved should be in sync with the option we see in the _Site Explorer_.

### What actually happened

The option gets out of sync.

As a result, the option in the UI is displayed as e.g. `Enabled` despite it being `Disabled` in reality (or vice-versa).

### Browser

_No response_

### Other information

_No response_

### Platform (Simple, Atomic, or both?)

Atomic, Self-hosted

### Reproducibility

Consistent

### Severity

None

### Available workarounds?

Yes, easy to implement

### Workaround details

If we wait some time and then try to adjust and save the site option again, it will get in sync again - unless we change and save it multiple times in a row again.

Reproduce the issue from Calypso Settings → Writing by changing rss_use_excerpt repeatedly, then compare it in Site Explorer on an Atomic or Jetpack-connected site. Trace the save and refresh flow for the site option; done means repeated consecutive changes remain synchronized between the Calypso UI and Site Explorer.

Written by the indexing model from the issue text.