This one to me does not mean 'to want someone to', but 'to want something done'. So I think, 'Todos queremos el problema se resuelva pronto' is enough.
If you have an introductory verb that triggers the subjunctive, like "querer" in "wanting something to happen", in Spanish the "que" is compulsory. You can drop it in English "I want something to happen" but we can't in Spanish.
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