Why is it impossible for a Quasi Star to exist these days?
This is a highly uncertain prospect, since the physics of the system itself relies on processes that are themselves uncertain to varying degrees, e.g., stellar nuclear evolution of very high-mass stars, accretion of gas onto compact objects, etc...
A relevant paper, cited by the wiki article for quasistars, is the work by Begelman $\it{et~al.}$ (2007). They show that newly formed quasistars should have surface temperatures higher than 10,000 K (9,700 °C). As a quasistar cools over time to a limiting temperature of 4,000 K (3,730 °C), for metal-free (Population III) opacities, at and below which hydrostatic equilibrium is no longer sustainable, the end of the quasistar's life leaves behind an intermediate-mass black hole.
Such quasistars would have a short lifespan, at most ~ 7 Myrs, during which the core black-hole would grow into an intermediate-mass black hole which are suspected to be the progenitors of current-epoch supermassive black holes. This implies that quasistars are transient, and so observing them directly would be very difficult. However, it is possible that we could observe indirect effects of their existence on their environments, for example this is how we infer the existence of common envelope evolution, or on the evolution of intermediate-mass black holes.