There's no functionality for that. You'd need that information sent to the substitution parsing API for it to be available for use.

I've no idea what a YOOtheme page is, but you should be able to do what you're wanting using a Code action in CB Auto Actions then set Method to HTML and output whatever page HTML structure you like. You can then link a Joomla menu item to that auto action using Community Builder > Plugin menu type. This effectively lets you build your own entirely custom pages with all the benefits of CB Auto Actions.

The page then does not convert the CB Substitutions code into the contents of the name field, but simply outputs the code above.
I thought that maybe reordering the plugins, putting ReReplacer before the CB System plugins would allow the changes to take effect before the page was rendered, but the output remains the same.

ReReplacer is a System plugin. CB substitutions are parsed by a Content plugin. So you won't be able to fix this issue due to load order. All content replacement plugins should be a Content plugin. ReReplacer has designed their usage incorrectly as a System plugin.