I've noticed a pattern with my stories. All of the times they start dripping with drama, someone is dying or about to die in an act of violence of some sort. It was pretty cool at the time, but now it's starting to become genuinely depressing (not really but you get my point).
That is a way to hook the reader. Another is a funny rather than a dramatic or dangerous. Season 2 of the Bakemonotagari anime is a case in point. Other stories happen right after the action in a crime scene. Others right before the confrontation. Others with foreshadowing. Try out different ones.
As long as there are big personal stakes involved, the story can be pretty dramatic. Life and Death is an easy way of going about it, but there are other ways, like... Problems within one's marriage, worry about passing entrance exams for university, depression, losing one's job and needing to pay bills and the like... Stuff like that can all be pretty dramatic~
How about the collapse of an empire? And not violently. Like the fall of the Berlin Wall. A situation where the rotting shackles of an old empire finally break and the new states formed now have to figure things out(And all the mistakes that entails). You could also look at it from the largest shard of the old empire, and see how the empty administration buildings and shell like cities are doing. The people, the legal and political rearrangements. The picking up of the pieces and gluing them back together.
Being in the midst of dying is merely one version of a "in the midst of losing something" style dramatic opening. Others include losing a dream, be it a marrage or other potential future (wealth, being disqualified for X job for reason Y, being fired from your job or expelled from school, losing a limb or functionality of a limb due to events, so on and so forth), a family member abandoning a child or the MC just getting bullied and/or mugged if you write it well. Remember, the point of a "dramatic" opening isn't for the story to be a drama, but to show that the MC is at a low point. It establishes a start point of "this is the reason why the audience can care" rapidly while getting us hyped to root for the MC. It doesn't need to be the lowest point the MC is at throughout the story, but it does need to be a low point. Of course, I am not a literary scholar, and am saying all of this based on my own conjecture and interpretation of the term "dramatic opening", particularly against the context of someone close to the MC dying. So I could easily be considered wrong, but I feel like this might be the advice you're looking for.
Amidst a conflict is quite interesting. Conflict with self, with others, with nature, something that would indeed show the character in a low-state. Another personal favorites of mine, is starting a story amidst confusion or a slow built-up of what seems to be a calm scene into one in the middle of chaos.
Part of (a) story formula is starting in the middle. The middle of dying, the middle of getting jettisoned 500 ft into the air. Personally I also Like to start 5 minutes before the middle, when the scientist is muttering to themselves "of course it's safe...perfectly safe" as their nose begins to itch and the dropper hovers perilously. Another part of the drama is knowing the impact of what's happening. Move back a bit if your story beats are missing a mark, like the deaths don't make the same impact or if you want an impact similar to death/dying, hook us on these things and show why loosing or being in these situations are important, or try other fishing gear. Writing styles change with time, so now that you've noticed a pattern that didn't bother you before, you now get to mess around and find out what fits you best now. As always, the answer is read more. Try a new genre, or old classics you never got to, you might find an answer there!
Try looking at real life stories and their drama. Random social drama from a petty topic, in work place. The reddit "r/" stories is wild sometimes, but take a grain of slat and make it compatible with your characters and story
What about, instead of death of a person, someone loses a lot of money? Maybe they are scammed? This is very depressing, and related to a lot of people. But the characters can still remain positive to try and get out of their situation.
I think is more of a case of making the reader understand why the situation is important to the character, and why he may not get it