NOTE: This worked for me as the issue came up with embedded dashboards in Experience Builder. Dashboards embedded via <iframe> elements are unfortunately still returning access issues.
It seems that the initial URLs used in the embed widget were either altered or are no longer valid.
Issue
In my case, the URL used in the embed widget that no longer works was the following:
https://www.arcgis.com/apps/dashboards/<item id>
Grabbing the dashboard URL from ago, the actual URL that does work is the following:
https://<organization>.maps.arcgis.com/apps/dashboards/<item id>
As seen, the previous URL was the generic, "arcigs.com/apps..." as opposed to the site specific, "<organization>.maps.arcgis.com/apps...".
Again, I don't know if this got changed in the update, or if the more generic URLs were acceptable for the embed widget pre-update.
Fix (works for embeds in widgets, but NOT for iframe elements)
To fix this, edit your experience and re-path the URLs used for the embed using the current URLs for the respective dashboards.
Amendment
One thing I did not test were iframe elements. My embedded dashboards were experiencing the same issue as you, however iframe embedded elements still return the permissions issue 😔