'Simpler' can be relative.
What I would 'simplify' in that tutorial is the use of the 'Up', 'Down', 'Left', and 'Right' words and replace them with actual angles. For example, the animation names would be: Walk0, Walk90, Walk-90, Walk180, etc. Then I would use the 8Direction.MovingAngle value directly and construct the animation name:
'Walk' & Player.8Direction.MovingAngle
That way I can avoid all the checks.