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No, that's not how backups work.

As explained in the documentation, Akeeba Backup will back up an entire site, files and database contents. Restoring the backup will result in an identical site, including the Joomla version.

It does not and cannot only backup the "contents" of your site. First of all, "contents" means radically different things for each site. Second, even if we used some very narrow definition (e.g. Joomla categories and articles, nothing less, nothing more) the way the core articles are stored varies greatly from one version of Joomla to another, they are intertwined with tags, history, permissions, user groups, access levels, third party plugin / component settings and so on. There's a separate class of extensions for content transfer (e.g. SP Transfer) but they do hit the aforementioned limitations. Moreover, if your definition of content encompasses anything beyond core Joomla articles you're outta luck. So let's not even discuss this method, this doesn't sound like something you could use.

Instead, what you should be doing is use Akeeba Backup to clone your Joomla 3 to a new location that has no version of Joomla installed. An empty site. That is to say, delete that empty Joomla 4 site and restore the backup of your Joomla 3 site in the now empty site.

Regarding the pre-update check, half of it is inaccurate and the other half is outright lies. As I have written in countless public tickets I had warned the Joomla project about this four times in the year prior to the Joomla 4.0 stable release. No action was taken 🤷🏽‍♂️ The end result is that some extensions which are compatible with Joomla 4 and do have a migration path are slandered by Joomla which states they are incompatible, whereas some extensions which are definitely not compatible with Joomla 4.2 are labelled as compatible because the extension developer said in their update site that this component is compatible with every PHP version from 3.0 to the (of course non-existent) 4.9999 — this is an impossibility, by the way, as there is little to no common Joomla core code which can be used across PHP 3.0 released in 2012 and PHP 4.2 released in 2022, a decade apart. So, whatever the pre-update check tells you take it with a grain (or a container ship full) of salt.

For our software, there are migration guides in our documentation. Here's the migration guide for Akeeba Backup: https://www.akeeba.com/documentation/akeeba-backup-joomla/migrate-j3-to-j4.html

For other third party extensions you should consult the documentation and support of their respective developer.

You will very likely need to use a different template than what you are currently using.

I recommend taking backups of your site as you are moving along and definitely just before and just after the upgrade from Joomla 3 to Joomla 4.

If the upgrade to Joomla 4 breaks your site: copy your backup archives somewhere safe, remove all files, then restore the latest backup you had taken using Kickstart. Please note that removing the Joomla 4 files before restoring the Joomla 3 site is very important. If you fail to do that the restored Joomla 3 site will be broken. This is explained in https://www.akeeba.com/documentation/akeeba-backup-joomla/restoring-backups.html#general-guidelines under “Do not try to overwrite one Joomla! version family with a different one”.