You continue to discuss the episode as if the story was about the Peter-Alex relationship. It was about the blood worms, (Doomsday biogenic weapons) and the issues about the blood worms, and the relationship was supposed to merely be a consequence of the main plot, as was the Klingon intervention which complicated it. The Klingons show that a crisis is not just a crisis, it effects other ongoing problem, and the death of Alex is no different than a lot of other deaths in TOS, just to demonstrate that the crew is vulnerable, and some people who aren't red shirts with targets on them get hurt and killed. Making him Peter's friend instead of James T.'s friend just avoided giving James T Kirk a new love to lose.
Ideally the show was about the bloodworms, but the choice was made to put the Peter-Alex relationship up front...it's so up front that the biggest scene about it (which is longer than most TOS teasers) comes before the real story actually starts. The relationship is then milked for melodrama.
Since I'm trying to be constructive rather than destructive, let me address how this could have been addressed. First, in the BAF Pt. 1 Teaser we see Peter get hurt, but since we've never seen Peter before, he's just another redshirt at that point. If fact, when he next appears, I didn't connect at first that he was the previously-nameless redshirt. Even if we didn't know who Alex was, if he's done something that illustrated his character, then once we find out who he is the preceding scene would have had some impact.
Back to showing v. telling, having Peter exhibit some characteristics or even charisma might have made him relatable and likable, and then there'd be a reason to care what happens. Instead, we're told he's a Kirk and he's in love, but absent him being an actual character those are just empty words.