I thought those solar panels are fake decorative elements only; the calculator still needs a battery to work and stops working when it runs out?

Some didn't. In today's world it is actually amazing how little power those things used. Both for that type of LCD screen and doing some calculations. Just compare it to almost anything else we use daily.

There were calculators in the '90s where the display would go faint when you covered the tiny solar panel. Perhaps the battery was already drained. Quite common I would say.

I had one in the 2010s when I was going through grade school. Casio fx-260 solar; it would die in seconds if you covered the panel.

The screen would go faint and slow when the power sources werent running at peak.

There's counterfeit ones yes, but somewhere I have a real solar calculator.

I even had a duplicate of it where I unplugged the broken solar panel and put 2x AA batteries on the back to power it. So the solar panel definitely worked