Is morphing a creature effectively a cost?
Turning a creature face up by paying its Morph cost is a Special Action, like playing a land.

Taking a special action is similar to paying a cost to cast a spell or activate an ability, in that both take effect immediately without waiting for anything on the stack to resolve, but there are also differences. Players can explicitly take special actions whenever they have priority, but costs can only be paid as part of taking some other action. So, any attempt to represent it as a cost would make it part of another action that does nothing except provide an opportunity to pay that cost, which is essentially backwards.

If Morph was represented as a cost to activate a nothing ability as described in the question, that action could not be taken while you cannot activate non-mana abilites because of cards like Overwhelming Splendor or abilities like Split second.