WOOO!!
I think your analysis is wonderful, you capture the little warrior so well.
If you allow me, I want to contribute a point of view that I have had for some time about Chay. I think that he has much more from Missa than can be seems.
Phil taught him to be a strong warrior.
Missa taught him to be a protective hero.
We might think that the conviction to care for Tallulah upon meeting her was driven by Phil, but that was born from Chay, before Phil told him. And that was due to Missa, who greatly shaped his personality from a very young age, strongly influencing what the boy would later polish with Phil.
The first days, Chay spent a lot of time with Missa and every opportunity he had, he repeated to him "You will be the hero of this town" "You will be the protector of what is good and just" "You are going to take care of me" "You will be strong to take care of everyone" "No one will be in danger while you are there" creating a hero complex in the boy (which also affected him negatively), so Chay promised to protect Tilin and also promised to protect JuanaFlippa (Chay and Missa literally promised it to Slime one day before she died).
The stories that Phil told Chay about Technoblade fueled his desire to be strong and powerful, to be a great legendary warrior like him and achieve many feats. Techno became his example to follow, but his conviction to protect others to the point of sacrificing himself if necessary, was born from Missa's kindness.
When Chay found out about the deaths of Tilin and JuanaFlippa (poor Trump, no one cared) he felt like he had failed. What kind of hero is he who lets those he swore to protect die? He was a fraud.
Phil didn't understand at first why this affected Chay so much, until he blamed himself, then Phil consoled him. But in this moment, Chay's protective personality was already well defined (Phil had not instilled it in him, Tallulah had just arrived, by that time, Phil had instilled more of that chaotic, adventurous, pyromaniac side to him).
In Tallulah, Chay saw the possibility of redeeming himself as a hero and without anyone telling him, he said was "I will protect Tallulah."
Phil accepted that, he saw it as a tender act on the part of his son and encouraged it and from then on, Chay became the protector of his little sister. For Phil it was fine, because Tallulah was fragile and needed a lot of care but he went to the point of putting that burden on his son. Chay took that job very seriously, he had to mature and leave behind many of his childish attitudes, because he was now the responsible older brother and that feeling grow for the other eggs.
When Missa returned (the first time) to Chay he could be seen acting like a child again, because it was just the two of them against the world. Chay had no other responsibility, he was only for his father and his father was only for him. For Missa Chay it was the only thing that existed and it had been a long time since Chay had been treated as an only child. That day he spent with Missa gave him the opportunity to relax and enjoy an afternoon having fun and making desserts, which helped birth his new Hobby "Cooking".
That same day, Chay thanked Missa for having been with him and said goodbye with the phrase "Thanks to you I am who I am", because Chay was always aware of that, he has always treasured his father's teachings and he loves him despite his absence.
Chay has also shown himself to be humble and selfless, wanting to give gifts to others and never envying material things. The seeds of that were the story that Phil told him "the enviuos egg" but that blossomed when Quackity stole his bike to give it to Tilin and Chay decided in the end to leave it to him (that didn't come entirely from Chay, on the way to Q's house , Missa was telling him that Tilin didn't have many things for having only one father and that maybe, if they had the bicycle, it was better that they keep it), Missa was willing to get it back if his son wanted it, but left the decision up to Chay hands. Being, in Missa's words, one of the moments in which he has felt most proud of him.
Phil and Missa's teachings were the perfect combination. While one taught him to fight in life, the other taught him to love it, in all its forms. Thus being the perfect balance.
Having been raised almost entirely by Phil, Chay takes a lot of his behavior and way of relating to others. He has his fighting techniques, his protection methods and understands his pranks. Chay knows him well and they are very close. But he always keeps in his heart, in an idealized way, all those loving moments that he lived with Missa, so much so as not to see his defects and mistakes as a father. With Missa he could open up emotionally, with Phil he didin't because he feels that he must be strong.
But not everything has been good. Because of the above, Chay has the worst qualities and anxieties of his parents as well. Acquiring Phil's paranoia about the dangers of the world they live in, which crushed his adventurous spirit and made him very careful about everything (since the nightmare). He is also suffering from the hero complex fueled by Missa, which makes him belittle himself and his own life, if it is for the good of others (just like Missa). Recently Chay also experienced abandonment anxiety, due to him feeling rejected by Missa after waiting so long for him.
All this makes Chay look like the most mature egg, but also the most tragic compared to the others, who have had more freedom (because as you say, Phil raised his children to be survivors), he is also one of the eggs who have experienced the most drama (Don't get me started on Tallulah, because she is the drama queen). He really needs Missa, to relax, have fun and enjoy life.
Of course, none of this is intentionally caused by his parents, they love him very much and Chay knows it.
I think I went on too long... sorry, this egg is the reason I became interested in QSMP and I feel that not everyone is able to appreciate him ( doesn't get enough love, at least from the Hispanic community). Whenever I see him, I see so much of Missa in him and somehow, seeing him in Phil's streams made me not suffer so much when Missa went to Japan.
Many people, especially those who did not see the beautiful relationship between Missa and Chay in the first days, exclude Missa from everything and consider Chay totally Phil's son, but those of us who followed him from the beginning know why, as you say, despite Chay loving Phil very much, Missa is his favorite father.
Thanks for read!! ♥
(I'm very happy for the return of my beloved boy.)
Sorry if there grammatical errors, english isn't my native language.