It's most likely that Coatlicue didn't have an equivalent predecessor. Please bear in mind that we aren't certain that the Sun had such a single progenitor star, but it's a very good hypothesis.
Star formation occurs through gravitational collapse in molecular clouds. As Wikipedia mentions, "the physics of molecular clouds is poorly understood and much debated", so there is an element of speculation in what follows. However, we do have quite reasonable models of the major processes that are consistent with the numerous observations that have been made of these stellar nurseries.
The fundamental process is Jeans instability, which
causes the collapse of interstellar gas clouds and subsequent star formation, named after James Jeans. It occurs when the internal gas pressure is not strong enough to prevent gravitational collapse of a region filled with matter.
The Jeans instability likely determines when star formation occurs in molecular clouds.
As the cloud collapses it tends to fragment, giving rise to stars of various sizes.
As you mentioned, our Sun's stellar nursery contained gas and dust from thousands of earlier stars (as well as primordial gas from the Big Bang).
As ProfRob said in the linked answer
The material spewed out from supernovae and stellar winds 5-12 billion years ago has had plenty of time to mix throughout the Galaxy before the Sun's birth. Turbulence and shear instabilities, driven by the winds and supernovae from massive stars, should distribute material on galactic-length scales in a billion years or less
Coatlicue was a large star which formed before the Sun, so it's likely that it was one of the earliest stars produced during the collapse process.
In this answer on stellar nurseries ProfRob says
The lifetimes of very massive stars (10-50 Myr) are comparable with the dispersion timescale for young clusters and associations. So massive stars tend to die near where they were born and they will be surrounded by their (more numerous) lower mass siblings. Lower mass stars have long lives and end up dispersing and making up the general population of a galaxy.
The early large stars affect the subsequent collapse process in their neighbourhood through their stellar winds and intense ionising radiation. Pela gives some information and diagrams in this answer on the "Pillars of Creation" in the Eagle Nebula.
So these large stars influence the formation of their siblings while they're still alive. And when they finally explode as a supernova the huge energy and matter release obviously has a big impact.
The formation of Coatlicue itself may have been influenced by winds and radiation from its large siblings, but because it was born early it's unlikely that its birth was triggered by a nearby supernova.
The key evidence for the Coatlicue hypothesis is related to radioactive aluminium-26 (Al-26), which has a half-life of 717,000 years. Aluminium-26 is primarily produced in core-collapse supernova explosions. As detailed in that Wikipedia article, H. C. Urey proposed the heat from Al-26 was highly likely to contribute to the known melting of small planetary bodies in the early Solar System. And if that's true, then there must have been a nearby supernova around the time of the formation of the Solar System.
Essentially all of that Al-26 has decayed by now to its stable daughter isotope magnesium-26 (Mg-26). The magnesium isotope ratios in meteorites are consistent with Urey's hypothesis.