Simple Sites: Site Editor will not save any changes to templates or template parts, multiple themes

Quick summary

I have a user with an FSE site that is refusing to save any changes to their Index template. No matter what I and the user do to confirm this change, it will not save at all -- not in the footer, or even in the main index content. It looks like it confirms saving, but once the browser refreshes, the change is removed from the editor and live site

Trying other blocks will not yield a saved change, either. I can replicate this repeatedly and consistently on their website, and both I and another HE could replicate this on our test sites using the Winkel theme, but not with any others thus far.

Steps to reproduce

Open the Site editor (Appearance > Editor [Beta]), and navigate to any template and Edit (Index, Home, etc)
Make any changes to the main content or to a template part (Header, Footer, etc.), such as adding a Paragraph block, etc. The change can be small, or multi-step.
Click on the "Save" button, and confirm the changes, then wait until the editor confirms the saved changes were applied.
Refresh the browser, and/or view the live site.

What you expected to happen

For the change to apply to the website with the given template after saving; and for the changes to appear in the editor view for a given template after a refresh.

What actually happened

The "saved" changes are undone, and cannot be found in either the editor or the live site -- as if nothing was changed at all.

Impact

One

Available workarounds?

No but the platform is still usable

Platform (Simple and/or Atomic)

Simple

Logs or notes

While this issue is only affecting one user, I and another HE can replicate this consistently so far for Winkel
I have tried multiple templates, multiple areas of the content, and multiple blocks, and the issue still persists.
This has only been tested on simple sites so far, both on a paid plan (Premium) and the free plan.
The issue makes FSE unusable with Winkel for this user, but I can advise them to switch to another theme in the meantime.
More details + a clear video of what happens on the user's site can be found in the notes for 5900185-zen, but the results should be replicable by following the above steps.

Automattic / wp-calypso