Yes, 'Peter sings better than they' sounds wrong to me. As far as I know 'them' is used when drawing comparisons (same with 'her' and 'him'. English has some oblique cases too!). "Peter sings better than they do" is grammatically correct, but to be really pedantic the structure is different than the given sentence (it has two verbs) so I think the given translation is the best.
"Peter sings better than they do", is less pedantic than saying "Peter sings better than they. " Both are correct English. Unfortunately, lazy English speakers say things like "better than them", and this is becoming the new norm. It is informal speech, but not accepted on standardized American university English tests as correct grammar. British tests of English seem to be more informal than American tests (First Certificate or Proficiency versus TOEFL). That said, Duo must accept the "correct" translations and not think that the informal way is the only correct way.
Both "better than them" and "better than they" are fine.
Some people think they're better than you because they say "better than I" instead of "better than me."
They're not, of course. They're just among the select group of grammar enthusiasts who think that than can only be a conjunction. You, on the hand, recognize that it can also be a preposition.
'Peter sings better than they' is also correct... Maybe a bit similar to answering, 'This is she,' if someone asks, 'May I speak to (Mary)?'
Interesting discussion on the subject. https://www.grammar-monster.com/lessons/than_I_me_than_he_him.htm