My unofficial generalized rule for ongoing has it suppress the state-based actions for phenomena and Sagas as well as schemes, since they are similar state-based actions.

I have designed ongoing phenomena myself, but there has yet to be a Saga design that actully would benefit from being ongoing (outside of counting as a card on the battlefield).

Well, it would be as easy as looping the chapters, right?

See: Year's Repeating Rhythm.

The mana ability of Urza's Saga can continue to work after it is finished if it is ongoing.

There are other possibilities too though, such as other abilities that care about the number of counters (possibly to remove them as a part of a cost).

Your example doesn't need ongoing (although being ongoing is harmless), since the counters will be removed before the state-based action can be applied anyways.

Now my generalized rule for ongoing includes dungeons as well (although since it says "enchant permanent", this card does not affect dungeons).

Ongoing also now (unofficially) affects battles; it suppresses rule 704.5v (a battle with no defense counters dies if it isn't the source of an ability that has triggered but not yet left the stack). Note that this does not prevent the intrinsic ability of Sieges from triggering, but it does prevent it from dying in this way if the intrinsic ability is countered.