Well, he's not too useful in an all-out melée, particularly when he's on his feet to dodge pikes.
When the hell would you not keep him mounted? He's mean't to be a cavalry hero.
His attack may be slow but he
can dish out damage eventually and he
can take a fair amount of hits. You don't need explosions and amazing splash damage to make a hero good. And let's not try to make Eomer the h-core melee damager either; If that were to happen, you would eliminate Eowyn's purpose.
Forgetting about Eomer in the late game is okay in my opinion. He has decent leadership abilities to benefit your army anyways. And also in my opinion, Eomer is one of the few heroes that you don't have to "babysit" to get the most out of him.
At the very most, I would only agree to an increased spear-throw range/damage when he reaches rank 7-10.
Not every hero needs a level 10 ability to be useful.
Eomer is just about perfect as he is in my opinion. You can get him early enough and he's a great economy boost, plus the leadership is excellent in a battle too.