nodes you output don't have type "node-set" - instead, they're what is called a "result tree fragment". You can store that to a variable, and you can use that variable to insert the fragment into output (or another variable) later on, but you cannot use XPath to query over it.
the xsl documentation https://www.w3.org/TR/xslt-10/#variables says:
Variables introduce an additional data-type into the expression language. This additional data type is called result tree fragment. A variable may be bound to a result tree fragment instead of one of the four basic XPath data-types (string, number, boolean, node-set). A result tree fragment represents a fragment of the result tree. A result tree fragment is treated equivalently to a node-set that contains just a single root node. However, the operations permitted on a result tree fragment are a subset of those permitted on a node-set. An operation is permitted on a result tree fragment only if that operation would be permitted on a string (the operation on the string may involve first converting the string to a number or boolean). In particular, it is not permitted to use the /, //, and [] operators on result tree fragments.
so using apply-templates on a variable doesn't work. this is actually where i got stuck before. i just was not sure because i could not verify that everything else was correct.
i wonder if it is possible to load the menu from a second document: https://www.w3.org/TR/xslt-10/#document
edit: it is!
<xsl:apply-templates select="document('nav-menu.xml')/menu">