According to The Best Answer above, it appears that you do qualify for the credit, but you may owe back some unemployment to your state if you do claim the credit. It would depend on your state though.
TurboTax does not ask if you received PUA because it does not disqualify you from taking the credit. However, if you do take the credit on your tax return, you may owe the state those payments back if taking the credit would have disqualified you from receiving the unemployment. @pumble
It will never hurt anything to file the extension.
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