Choosing between oblivion, forgetfulness, and forgetting

Oblivion might work, as long as the connotations line up with the Spanish olvidar in a way that satisfies your sensibilities as a translator. Otherwise, forgetting is laden with fewer negative connotations and may be a suitable literal term.

Oblivion

The issue with oblivion in English is that some of its meanings have negative connotations. For instance, the Oxford English Dictionary defines oblivion definition 1a and 1b as:

1a. The state or fact of forgetting or having forgotten; forgetfulness; (also) freedom from care or worry.

1b. Forgetfulness resulting from inattention or carelessness; heedlessness, disregard.

1b poses a problem. Oblivion can be seen as a kind of carelessness or inattention, which goes beyond memories fading with time. That is not to mention the destructive connotation:

2a. The state or condition of being forgotten; (also, more generally) obscurity, nothingness, void, death.

The idea of falling into oblivion, that is, dying or being entirely forgot, is a part of the word. So you would want to make sure that olvidar has similar connotations or that you're okay with your translation having that be a possible reading.

Other possibilities

Forgetfulness runs into a similar issue that oblivion did; it may suggest a negative mental capacity or a negative moral quality (see definition 1b above). In my experience, that sense is even stronger with forgetfulness than with oblivion. Definition 4 of the word gives a particularly strong example:

4. Disregard, inattention, neglect.

1757 S. Johnson Rambler No. 180. ⁋5 He..naturally sinks from omission to forgetfulness of social duties.

For that reason, I would suggest avoiding it.

Forgetting is an attractive word to use in this context. It is close to the verb and hence closer to the basic sense to forget, with fewer negative connotations. It also has precedent in William Wordsworth's Ode on Intimations of Immortality:

Our birth is but a sleep and a forgetting;

The Soul that rises with us, our life's Star,

Hath had elsewhere its setting

And cometh from afar [...]

So poetry can combine forms that might be odd in other contexts: "a sleep and a forgetting" or "forgetting and hope."

• 'oblivion' is a very strong negative, and is rarer than the other two, and is a much more academic word than the others. It means a total eradication from everyone's memory. It is often associated with total destruction or annihilation, to the extent that no one remembers the thing destroyed any more.
• 'forgetting' is a past participle or gerund, so it is very verb like. It is not common as a gerund (noun). Note that it doesn't have its own dictionary entry, coming under 'forget', but can still stand alone as the abstract noun for 'the process of losing memories'.
• 'forgetfulness' is half suffixes, is common, and often taken to be a symptom of old-age or a deficiency.

'Oblivion' is not recommended even if it is the accepted translation of 'el olvido', since 'oblivion' is so extreme in English. Is 'totally forgetting everything about something' really a 'force that helps us live'? That seems a little weird to pair with hope and also you don't often think of oblivion (total loss of a memories of a thing) to be a positive. Maybe just the ability to forget little would be enough.

So in a pair along with 'hope', the most parallel would be 'forgetfulness', but 'forgetting' may also be appropriate since its association is just the process and not a slightly negative

I take the quote to mean that what drives us forward in life, is both the promise of something better (hope), and the forgetting of suffering and other evidence to the contrary.
In that last sense it feels slightly analogous to the proverb "ignorance is bliss", but ignorance doesn't seem to come close to the intended meaning of Blasco Ibáñez.

I think of the three forgetting is the best option:

• Forgetfulness implies a more active loss of memory. It is only really used when someone has a relatively higher tendency to forget things.

• Oblivion is a state of being completely forgotten. It is what has happened to most people throughout our history (and, necessarily, before that), so certainly not an impetus to live.

• Forgetting comes closest, and, although it might not be sufficiently precise as anything can be forgotten, for the sake of lucidity (of the aphorism) this seems the best option.

Between your three options, I would rank them forgetting best, oblivion worst. If you're willing to use a phrase rather than single word, you could say "impermanence of memory" or "imperfect memory". There's also "amnesia", but that implies a state of being more forgetful than normal, and this saying seems to be saying that normal human forgetfulness is helpful.