Could population III stars develop directly into population I stars?
No they couldn't. Population I stars contain elements like strontium, barium, gold, lead etc. that are not formed (much) in type II (core collapse) supernovae. They are also much richer in iron, nickel, manganese etc. than are stars formed from gas enriched only by supernovae.

Population I stars (like the Sun) can be termed third generation stars because they contain material that has been through at least two stars - but not just two high mass stars that exploded as supernovae.

Iron-peak elements are mainly formed and disseminated by type Ia supernova that are exploding white dwarfs. White dwarfs are the remnants of low-mass stars with long lives.

Many heavy elements (strontium, barium, lead etc) are formed by the s-process in intermediate mass stars that also have (relatively) long lives and do not explode as supernovae. Furthermore, this neutron capture requires iron-peak nuclei to act as seeds, so these stars must in turn have formed out of material already enriched with iron-peak nuclei.

It is also now thought that other elements (silver, gold, osmium etc) are formed by the merger of neutron stars, where there must be a significant delay between the initial supernovae that produced the neutron stars and the subsequent merger by orbital decay.