In hypothetical "if-then" statements, the "if" part of the sentence gets the imperfect subjunctive, and the "then" gets the conditional. (In this sentence, there's no "then" clause, so we only have the "if" clause to worry about, and that gets the imperfect subjunctive.)
In case it's any help, I always think of this as the "if I were...." rule. When we talk in hypotheticals in English, the "if" clause gets what USUALLY looks like an ordinary past tense: "if I wanted a blue car, I would have one." But actually, it isn't an ordinary past tense, because when the verb is "to be," the correct form is always "were" (even with subjects that would normally use "was"): "if I were you, I would buy a blue car." So what that verb form actually is is our equivalent of the imperfect subjunctive. (For the longest time, I used to have to remind myself, "if it's an 'if I were...'-type sentence, it gets that weird subjunctive form that I can't conjugate without looking up!" =) )
It works as a complete thought in Spanish, and we're giving some consideration to the English translation, here. In English (for reasons I can't begin to figure out! =) ), it works as a complete sentence if we insert an "only": "if only I could start again." If we don't, it doesn't.
My suspicion is that it would be better to use "if only" in the primary translation, but since the sentence has to be used in a couple of different ways (and also in the "English for Spanish speakers" tree), I want to think a little about the implications of adding an untranslatable "only."
You're definitely right about the unstated clause; I just get the sense that the "unstatedness" is more comfortable for English speakers if there's an "only." If I say, "if only I could start again," people nod sympathetically. If I say, "if I could start again," they look up expectantly: if I could start again, then what?
We have a handful of sentences like this, and the primary translations haven't included "only," but they were generating a certain number of confused and/or upset error reports from the English side, because a lot of people thought they felt too incomplete. After some discussion within the team, we're test-running "if only" in the primary translations, and we'll see if people find that easier, or if they just find it confusing in a different way.
I'm not sure it's possible on Duolingo. The Incubator has to check and approve words and punctuation as we type them in (this is so we don't use words that you haven't been taught or punctuation that the program doesn't know how to handle), and unfortunately, it doesn't seem to recognize ellipses.
The "only" gives it the sense of a wish (As does the older use of "but"). You are stating a condition contrary to fact: " If only I could fly like a bird" It might have an unstatef cause, but it might also be something un-stateable. Much like when I have heard Spanish speakers start with "Ojalá"
"If only I could begin again" is a wish for the future, "if only I could have begun again" is a complaint about the past. https://dictionary.reverso.net/english-spanish/%C3%AFf%20only%20I%20could%20have%20begun%20again
"could" is a timeless modal that can be used for past, present and future. Some dictionaries list it as the form it was originally formed from which is the past of "can", but it has far exceeded that use and is used in conditional and subjunctive forms now. This is a subjunctive form, a wish for the future, so the past would not work here. The Spanish is more specific and it is not a past version here. So although "could" can possibly mean "was able to", "pudiera" is a subjunctive preterito imperfecto. I wonder if they would allow "If only I were able to begin again" ? http://conjugator.reverso.net/conjugation-spanish-verb-poder.html
What I imagined when I read the Spanish sentence is a speaker who started a talk, was interrupted, and dealt with the interruption. I can just hear such a speaker then say: "If I could begin again..." and launch off into his/her talk. I expect I HAVE heard such a thing in a class at some time.