Yes, but it took the Pop III stars to allow those smaller stars to pop into existence.
What is the evidence for this? None. Smaller stars have been forming since the cosmic dawn, and cranking out lots of heavy elements by SNs. And they are still forming today, without Pop III stars.
SN 1987A rewrote the book on core collapse SNs. Before, they were thought only to occur from old red giants. That a star only 12 million years old could collapse clearly proves that early heavy elements do not need supermassive Pop III stars for their formation.
Pop III stars are a hypothesis to explain the existence of heavy metals in the early universe, and some suggest their black holes merged early to form SMBHs. There is no evidence for any of this, so it is not a fact from which to draw the conclusions you are making.
Current astrophysics tell us that the maximum mass for a stable star is about 150 SMs. Anything much larger will not persist for very long, or even form a star. Such a mass might even undergo direct collapse to a black hole without a SN.
en.wikipedia.org
Is this the logic for expecting very old red dwarfs?
The logic for very old red dwarfs is their small size, and their very slow fusion rate. Their elemental composition could have been provided by standard SNs like we know of today.
They are expected to last for potentially trillions of years, and have almost certainly been forming for billions of years, just like all the other stars that we know are real.
Again, none of this relies on those hypothetical Pop III stars.