No, it does not. I do test before releasing and this is one of the things that get tested every time.
Just out of curiosity, I installed a brand new Joomla 4 site and installed Akeeba Backup 9.4.4 on it. I went to Components, Akeeba Backup for Joomla, Manage Backups. Got the popup. Click Got It. Page reloads, no popup. Went back to Joomla's control panel, then Components, Akeeba Backup for Joomla, Manage Backups. No popup.
If you still get the popup it means that clicking the button didn't actually access the URL that hides the modal on your server (index.php?option=com_akeebabackup&view=Manage&task=hidemodal), it only had the browser load the redirection from that page (index.php?option=com_akeebabackup&view=Manage). As a result, the flag to hide the modal was never recorded. Try using a different browser.
Another possibility is that your sites are behind a CDN (e.g. CloudFlare) BUT you have not told the CDN to not cache the pages whose URL starts with /administrator/index.php. Or you are using a host-provided cache (e.g. LiteSpeed Cache on Rochen, SuperCache on SiteGround, etc) which also cache these URLs. Fix the cache.
Please remember. We have no control over your browser, CDN or cache configuration. If either is configured to make it so that a URL is never accessed on the server we have no way to work around that; it's under your control, not ours.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!