Adding a little something to this, because I feel like I need to, be welcome to disagree.
"antique Roman" has multiple meanings here, each and every one gayer than the other. You already unpacked the socially accepted homoerotic content, so I wouldn't get into that.
Killing oneself, though, was something that wasn't too unfamiliar in Roman culture as well. We can think of Anthony and Cleopatra, Nero, Cato, or any other infamous case, when love or values were stronger than the desire to live. Horatio basically strengthens his statement by saying "You are more important to me than life, therefore I will die, whether you like it or not".
Another aspect worth mentioning is the Renaissance perspective on suicide. Back then, people were defined by gender roles, so much so that the way you died could be masculine or feminine too. This is something Shakespeare loves playing with, let's take good old Romeo and Juliet as an example, where Romeo, the male counterpart ends up drinking poison, generally believed to be a feminine death, and Juliet stabs herself which is a masculine, heroic way to die.
Let's see what happens in Hamlet. We know that Hamlet has a dagger, he plays around with it in his soliloquy, and Horatio could have easily grabbed it to stab himself. Ooo, some person in the audience says, but they were in a fencing match and Hamlet stripped his unnecessary clothing! Yes, and gave it to Horatio. See what I mean?
Instead, Horatio reaches for the cup. He is a scholar, well aware, that drinking poison would be a, and now I use these words to emphasize my point, "coward, pussy way to die". He is still willing to do it, because he loves Hamlet so much he doesn't care how people would remember him.
Ooo, but he didn't consider that, some would say, and you are welcome to do so, because okay, let's say he didn't, but you know who did? Good old William Shakespeare, who wrote basically NOTHING without intention, mostly not such a drastic scene. Romeo and Juliet didn't consider it either, just reached for the closest thing available. Yet, to the audience, it gives a message, that Romeo was just a kid, and that Juliet took a weight too big on her shoulders. Horatio, ever the anchor of Hamlet is called manly by the prince multiple times, who's unsure about his own manhood, but this is just a side note. This Horatio dying by poison would give the idea of a soft, loving, even domestic person, which are rather unmanly traits at the time.
This highlights the potential queer background even more. Horatio is too gay to die in a manly way, this is the message we get through this scene.
