There should not be any notifications, so there should not be any “flashing” of notifications. You may not care about the organization logo, but Google certainly does.
Assuming the featured image is large enough – ie. more than 1200px in width and height – then WPSSO Core will use it and there should not be any notifications about a missing image (as WPSSO Core will be able to use the image). If the image is too small, then WPSSO Core will not be able to use it and will let you know. There should be several suggestions to provide a better image in the first notification message, but that message is dismissable and you must have dismissed it (as it is not showing in the screenshot you provided).
Is your featured image larger than 1200x1200px? If not, then see here why you should not upload small image to the WP library: https://wpsso.com/docs/plugins/wpsso/faqs/why-shouldnt-i-upload-small-images-to-the-media-library/. You can also select a larger image in the Document SSO metabox when editing a post/page.
Note that the SSO > Advanced Settings > Plugin Settings > Integration > Image Dimension Checks option is now unchecked in the WPSSO Core Standard version, so smaller image may be included in your markup (and thus not show a notification about missing images), but social and search engine validation tools may give an error if the image used for that markup as being too small (ie. less than 1200px). 1200x1200px really is the smallest image that should be uploaded to the WP media library – a 1200x1200px image only fills a 600x600px container for most devices (as most devices now have high resolution screens that have twice the pixel density).
js.