Can you provide a test script and its actual vs. expected output to show exactly what you mean?

<?php
$xml_header = "<?xml version='1.0' encoding='utf-8'?><xml/>";

$xml = new SimpleXMLElement($xml_header);
$xml->addChild("first_string","this is &lt;mystring&gt;");
$xml->addChild("second_string","this is &lt; mystring&gt;");
$xml->asXML("test.xml");
echo "<pre>" . $xml->first_string . "<br />";	// 'this is '
echo $xml->second_string . "</pre>";			// 'this is < mystring>'
?>

Shouldn't the values passed to xmlNewChild in addChild go through BAD_CAST like 
all other Xml related methods do?

Something is wrong here.
The tag is not removed, it's not encoded anymore. But &entity; are removed.
See  http://3v4l.org/EJGuL

> Something is wrong here.

No, everything works as expected, albeit the behavior is not
documented.  Firstly, I doubt that the tag has ever been removed
as reported originally; almost certainly the tag was simply not
shown by the browser.

Anyhow, SimpleXMLElement::addChild() uses xmlNewChild() under the
hood, and the relevant documentation[1] states:

| If @content is non NULL, a child list containing the TEXTs and
| ENTITY_REFs node will be created. NOTE: @content is supposed to be
| a piece of XML CDATA, so it allows entity references. XML special
| chars must be escaped first by using xmlEncodeEntitiesReentrant(),
| or xmlNewTextChild() should be used.

When SimpleXMLElement::__toString() is called, it uses
xmlNodeListGetString() under the hood, whose documentation[2]
states:

| Build the string equivalent to the text contained in the Node
| list made of TEXTs and ENTITY_REFs

So *known* entity refs are resolved, while unresolvable entity
refs are skipped (€ is not predefined).

Note that creating empty child nodes, and setting their value
afterwards via assignment has different results[3], because the
assignment applies xmlEncodeEntitiesReentrant() automatically.

Changing to documentation issue.

[1] <http://www.xmlsoft.org/html/libxml-tree.html#xmlNewChild>
[2] <http://www.xmlsoft.org/html/libxml-tree.html#xmlNodeListGetString>
[3] <https://3v4l.org/JMd3W>