The problem is, the star would "literally" blow apart
The problem is, modern stars tend to be way too small to even hold themselves together if the core collapses into a black hole.
For starters, a Thorne-Zytkow object (called "hybrid stars" by astronomy rookies) is basically in simple terms, just a red giant/red supergiant with a neutron star at its core, which is formed when a neutron star collides with a red supergiant, most likely occuring in a binary pair.
Quasi-stars/black-hole stars on the other hand have been theorized to have been somewhere between $10^3$ to $10^7$ solar masses. Meaning that a quasi-star would have had to be about as massive as thousand to 10 million solar masses. In comparison, the most massive star today is R136a1, with an estimated 250-300 solar masses.
The reason why a Thorne-Zytkow object cannot form a modern Quasi-star, is basically because, black holes are frigging efficient and converting mass to energy. According to this paper from Stanford University, black holes convert 40% of the mass that falls into them as energy. In comparison, nuclear fusion has a measly 0.7% mass-to-energy conversion rate.
This means that, even if the neutron star inside the TZO collapsed into a black hole, the star would last for a very SHORT time. Like, literally, you only have seconds or minutes left (if you are lucky) before the energy and jets from the black hole formed would RIP the star apart into pieces. It would be the equivalent of putting the Unicorn black hole, inside Arcturus/Betelguese (either case, both stars explode from energy produced).
Thus, my short answer is: "No, a Thorne-Żytkow object cannot be the progenitor of a modern quasi-star".