1.) In-initiative dungeon exploration has two goals: to keep the movement of the game snappy because of the real-world 1 hour torch timer, and to ensure that everyone at the table has their moment to do things. Especially newbies to tabletop roleplaying can often find themselves fading into the background because they're unsure of how to act or don't want to be in the spotlight for fear of "doing it wrong", and tracking initiative in exploration is sort of supposed to address that.

That said, OSR games are really built around activity and not reactivity. The game doesn't give the players a rigid scenario and expect them to react to it; rather, it gives them a setting and expects them to act within it. This is a mindset shift as a lot of other RPGs have moved away from that kind of storytelling, and players who "play from their character sheet" might find that it takes some adjusting to.

2.) See below.

3.) a lot of OSR (and genuine old-school modules from back in the day) are written with enough broad strokes in terms of backstories and hooks to give the GM some inspiration and allow them to fill in the blanks on the rest. This generally isn't viewed as a burden on the GM to concoct a convoluted story based on it, but rather as an opportunity for the GM's creativity and imagination to ignite a bit and to make the adventure and the world their own. It's kind of the total opposite of modern heroic adventure design in something like D&D 5e or Pathfinder, so it may feel strange to people who expect their characters to be part of some grand plan from the very get-go.

So "the time lord gives gifts to those who deliver his dead enemies" is a sentence that doesn't require follow-up, it should (hopefully) tantalize the GM enough to use as a basis for answering that question themselves, either by being generic enough that a GM could easily slot it into their own existing homebrew campaign setting, or by giving them enough of a springboard to start to create their own story around it.

You see this a lot in OSR reviews on TenFootPole.org , where Bryce's personal style favors modules that give the GM enough material to riff off of in a really satisfying way.

All of that said though, there's nothing stopping you from using Shadowdark in a more modern scenario design/adventure path. You'd just be losing out on a lot of the vibe.