UPDATE & SUMMARY (as of 4.3.0):
For everyone having issues with the Kodo stopping at the first group of mobs & hanging there.
-despite AngryLittleBunny's post, there is no Kadrak in Splintertree until AFTER you complete the quest, & he will accept the quest hand-in fine.
However, as for actually getting there:
Almost nothng in this thread worked, & what did work was very specific:
-you can't hearth off the kodo, as you get an on-screen message saying "You can't do that right now";
-ditto for the 'Character Stuck?' option mentioned by Taake;
-you can't, as Arearda mentioned, simply take the quest then "follow along" with the Kodo or make your own way to Splintertree;
-logging out and logging back in will not fix the problem alone.
Here is how it worked for me:
-take and begin the quest;
-get stuck;
-DO NOT ABANDON THE QUEST (nothing else here will work if you simply start all over again);
-exit the game entirely (ie, don't simply log to Character Select);
-log back in; you will be at the place the kodo got stuck, but you will not be mounted;
-hearth out of the zone (if you are bound in Ashenvale, this will not work - change your hearth point & try again);
-if you have a Haunted Memento, bank it for the time-being;
-remove any gear (including weapons - you will not need them on this quest anyway) that has animated visual effects (putting them in your bags will be fine);
-fly back to the Mor'shan Rampart, but make sure you dismount a fair way out, and run in on foot;
-talk to Kadrak again (you should still have the quest from before) to remount, and voila! the quest should complete fine this time.
TheWoWDB
