If they don't want to play a cleric that's fine. The party will be fine without.
But, in terms of what makes them actually good in 5e I would say that it is:
1. They get the most spells prepared because they get as many as a Druid or Wizard and then also get a bunch of Domain spells. The Domain spells tend to be a mixed bag, but there's always something good. Other than a Wizard who really heavily invested in learning ritual spells, or some crazy multiclass, they will have the most spell options at the ready at any time.
2. Half the classes can heal, but clerics are still the one stop shop for the whole panoply of restorative magics. Lesser restoration, greater restoration, remove curse, and all sorts of dead raising. These are where being a prepared spellcaster really shines as they are spells which are probably not needed on a regular basis but are periodically of vital importance. If the group doesn't have a cleric it is fairly likely that at some point they will get some disease, curse, petrification, or whatever that nobody has access to the spell to fix. This is a good opportunity for the DM to send them questing after someone who can solve the problem, so, once again, maybe it's not so bad if none of them want to be the Cleric.
3. With revivify they can raise the dead at level 5. Everyone else who can raise the dead (some subclasses excepted) needs to wait until level 9. That's a substantial space in the levels where people actually play a lot where Clerics still have a serious monopoly.
4. Your group is right that Paladins and Bards are both pretty awesome in 5e, but neither is quite a replacement for the Cleric. Paladins know similar spells to Clerics, but Paladin spell progression is just too slow to compete. Bards have access to some great Cleric spells, but many of the Cleric, Druid, and Wizard spells they get are mismatched to someone who isn't a prepared spell caster. Occasionally great, but day to day unnecessary spells like Zone of Truth or Speak with Dead are probably not going to make the Bard cut unless they are particularly thematic to the character, whereas any Cleric might choose to prepare them on a day where they seem likely to be useful, and potentially really important things like Greater Restoration or Raise Dead feel like a tax on the Bard's spells know that they pay for not having someone else in the party to handle them.