I’m not sure if this will answer your question or not..
WordPress separates the Content (pages, posts, media etc) from the design and functionality through the Theme. So you can completely change the design and functionality by changing the Theme, but retain the same content. You can see this in action with a new WP install by switching between TwentyTwenty and TwentyNineteen themes for example.
The reason for mentioning this is that you can have a completely ‘clean start’ by creating a blank custom theme and switching to it. This will dispense with any poorly built templates etc but means you are starting from scratch (though this may be what you want).
So with that in mind, one potential approach is to take a complete backup of the current site, migrate it to your preferred dev platform (e.g. WAMP/MAMP) and create a completely new Theme container. Switch to that Theme and you have a blank canvas but have retained all of your content, plugins, core config, user/subscribers etc. No messing with export/import utilities which as you say, miss vital stuff. You can even keep a copy of the old Theme (deactivated) for reference as to how the previous developer did stuff.
In terms of how you migrate the site, this is possible via manual backups of both the DB and entire WordPress directory but there is a need to run specific processes against the SQL in order to update URLs throughout. However, there are free and reliable plugins for this task, so please do a bit of research for migration/duplication plugins.
To create a bare bones theme from scratch, read up here