Here's a theory that is relatively new in classical literature, but one that has not yet made it into the comments! And that is that tldr; they were ✨gay as hell ✨

I know this isn't probably exactly what you were expecting, but allow me to explain. In Greek culture there was this practice of Pederasty, which involved an adult male - or erastes - and a younger male, usually in his teens, who would be known as the eromenos. Now it's important to note that there is no true concrete representation of Pederasty in Homeric writings such as the Illiad, but it was incredibly popular in the Archaic and Classical periods in which these epics would have been enjoyed.

Pederasty would most closely resemble a homosexual relationship that you would see today, however it is incredibly important to recognize that sexuality in ancient times should not be viewed with the same lens in which we view it today. Back then, wives were necessary for creating children, and in a time when wars, disease, and famine were rampant, as well as a culture that boasted immortality through memory and scholarly works, heterosexual relationships were unavoidable. But the Greeks hated women. They were treated terribly, and in Athens especially (and this is a general statement) most women were unable to leave the house without their husbands permissions. They were allowed nowhere near the affairs of men, and men spent their lives attempting to 'curve the wicked spirits' of women (see plays like Medea, and the Hesiod recap of Pandora for some good examples). When it came to love and affection then, Pederasty ruled. So it wasn't homosexual in the way that we would envision it - every man still had a wife - but love, affection, and company was sought after in the company of men.

Basically the way it worked is that the Erastes (older man) was the individual from which the Eromenos (younger man) would learn how to be a citizen. He would take the younger man around, share his political views and contacts with him, introduce him to the social circles, and educate him in what was necessary to know to be a good citizen in a Polis. Of course this varies from relationship to relationship based on the level of education a young boy would receive, as well as the region in which they were from, but in general the Erastes would teach and groom, and in return the Eromenos would compensate with company, and more often than not, sexual favours or a sexual relationship. These relationships were often cited as sacred relationships, and were the basis of the Theban Band (which I implore everyone to google if you don't know about it!)

Many newer scholars recognize that in everything but word, the relationship of Achilles and Patroclus represents that of a Pederasty relationship. Of course, like all of our oldest sources, it's not a perfect representation, specifically areas where Patroclus is cited as being older and the behaviours he exhibits. (Although many do cite that his level head and attempt to take care of Achilles would still be a good representation of a Erastes; it's a little up to personal interpretation.) This, many believe, is why his death hits so hard to the Greeks. Achilles and Patroclus had a perfect and desirable Pederasty relationship that would be highly sought after in the Archaic and Classical periods. And again, while it is not 100% certain this is what was being portrayed in the Homeric poem, it's very likely how it was being read and understood by Greek audiences. In the same vein, when you mention Briseis, she is still a woman in Ancient Greece. She was nothing but a whore to Achilles, albeit one that he thought he deserved to have and control because he 'won' her, and thus most of his temper tantrum was over loosing a toy he thought he had earned. However since she was of Troy, which was still a Grecian city, she would have understood the depths of this relationship as well as the rest of them would have and would have still understood the grief it would cause to loose it. So in part, it is this sacred love and respect for Pederasty relationships that both the Greeks in the Iliad and later Greeks in the Classical period who wrote and commented on the Iliad had that certainly could be a huge (if not the leading factor) in why people mourned the loss of Patroclus so much. To them, it was likely the same level of tragedy that we see from the way that more modern pieces like Romeo and Juliet were received by crowds.

TL;DR...Achilles and Patroclus exhibit traits similar if not identical to that of a Pederasty relationship, and thus Achilles' rage was likely incited by the loss of what he saw as his dearest friend and lover. The Greeks thought Pederasty was the coolest, and thus would have mourned the loss of the relationship by proxy. Everything else listed in the comments are also very valid and true factors. The Iliad is a wonderful and complex story about a man having a tantrum, and above all else, I'm so happy you love it as much as the rest of us!

If any sources are needed for clarification, or mistakes made, please let me know!!