How do you revise/modify a single value of an XML element with Extendscript?

I stripped your code down to bare essentials. I was able to insert dateRun into the XML without any problems. Here's my code.

function readXML(toRead){
  toRead.open('r');
  var xmlStr = toRead.read();
  toRead.close();
  return new XML(xmlStr);
};

var theFile = File(Folder.userData + "/Image Processor Pro.xml");
// Assume the file exists

var xmlReadObj = readXML(theFile);
$.writeln(xmlReadObj.toXMLString());

xmlReadObj.dateRun = "test";
$.writeln(xmlReadObj.toXMLString());

Note that you'll have to have ESTK up and running to see the $.writeln output.

You do need to write xmlReadObj.toXMLString() to file to update it with the new value. Here's how I do it in xtools/xlib/stdlib.js:

Stdlib.writeXMLFile = function(fptr, xml) {
  var rc;
  if (!(xml instanceof XML)) {
    Error.runtimeError(19, "xml"); // "Bad XML parameter";
  }
  var file = Stdlib.convertFptr(fptr);
  file.encoding = "UTF8";
  rc = file.open("w", "TEXT", "????");
  if (!rc && Stdlib.IOEXCEPTIONS_ENABLED) {
    Error.runtimeError(Stdlib.IO_ERROR_CODE, Stdlib.fileError(file));
  }
  // unicode signature, this is UTF16 but will convert to UTF8 "EF BB BF"
  // optional
  //file.write("\uFEFF");
  file.lineFeed = "unix";
  file.writeln('<?xml version="1.0" encoding="utf-8"?>');
  rc = file.write(xml.toXMLString());
  if (!rc && Stdlib.IOEXCEPTIONS_ENABLED) {
    Error.runtimeError(Stdlib.IO_ERROR_CODE, Stdlib.fileError(file));
  }
  rc = file.close();
  if (!rc && Stdlib.IOEXCEPTIONS_ENABLED) {
    Error.runtimeError(Stdlib.IO_ERROR_CODE, Stdlib.fileError(file));
  }
  return file;
};

The error handling code is in there because it's easier to track down IO problems when they occur.