There might not be good way? But you don't really want to do this. To say "here is a thing, you must do it to play the game" is the wrong way to go.In contrast to "only you can stop the villain and if you don't try there would be no game", what available methods do we have to attract PCs to make someone else's problem their own even though the campaign does not require them to?
You can take as long as you need. Why do you feel there is a rush of just one week?If you have an existing group of players and they are asking to prepare and run a new campaign for them, then you can ask them to decide on the pitch and come back to them a few weeks later. But of you want to pitch a campaign and go looking for players, the game needs to be ready to start within the week at the latest.
For new players, new players to my group or players that "don't know what they want to do" I will start with a forced plot. the tavern explodes, the PCs fall into a 20 feet crater and some dragonborn are like "quick finish them off before they fulfill the prophecy!" Or Aunt Voras shows up with a Ring of Daily Wishes, and asks the PCs to hide it, but not use it.....want to GUESS what happens EVERY time I do this plot?If not hooks to reel them in, there probably need to be at least some ropes within grasp that the players can pull on to see where they go. At the very start of the campaign it's often best to have the characters in a situation with very limited options and a immediately pressing problem that must be addressed. That gives them some time to get familiar with the environment and a feel for who their characters actually are. But while that helps to get things going, it still only pushes out the moment where the PCs will have to be thrown into the water to swim or sink.
A DMs wold building is a whole other topic. A lot of DMs like simple, classic , low fantasy, low magic, low adventure type worlds. So there might be a lot of "just like Old Tyme Earth" plots and stories, but there won't be much like "an amazing fantasy adventure" plots and stories. This is the typical disconnect that is seen between many DMs and players: The DM wants a game like the movies Excaliber or Brave heart, the players want a game like Big Trouble in Little China or Harry Potter.Many game concepts have PCs take on jobs or other tasks for rewards, but it always feels to me that these rarely ever sound actually exciting. They are stuff you accept to do because you understand the game needs you to do so. Again, in the introduction to a campaign, that's not a bad thing once or twice. But if the players/PCs are supposed to become proactive at some point, the world needs to have things that actively sound attractive and exciting. Things that make the players want to drop the safe and predictable options and instead take avoidable risks.
Well, player investment works. Give the players real things in the game to earn. Ones I have used:I think that's kind of the goal I am after. Players having every option to just walk away from a situation and keep their PCs safe, but still rather wanting to do that thing that might cost them dearly. Which is why I feel it is very important that the players pick for themselves what they want to get invested in. If it is part of the premise of the campaign that the PCs will be heroes for a specific cause, then the players know that walking away is not actually a real option that they could go with and continue the campiagn.
*Have the PCs be super powerful creatures, like dragons or Demon lords...EXCEPT...they were defeated in battle and now are shadows of themselves. So, they make 1st level characters, but have some powers or such. What they have to do is find the three/six/ten/whatever 'pieces' of themselves to make them whole.
*The PCs start as undead, like ghosts, with no memory of how they died, but lots of clues that they were a powerful group that ruled a empire or such.
*Dark Matter(the TV show): The PCs wake up on a Spelljammer with no memories of who they are....but there are plots around them. A fun twist here is have the players each make a character, then randomly switch them. Even more fun is the DM keeps all the sheets until the players figure out who and what their character is.
*The Lost Kingdom: The PCs are the last royalty or such of a kingdom that was over run, destroyed, or such....and they have to rebuild it. A real fun twist here is the PCs are out of time...like they were in timeless stasis for 100 years or more...
*In Time- the PCs are all ancient warforged or such awakened in the modern day...but with 'low batteries'. They need to absorb magic to live, something like 1000 gp equals 24 hours of life or such. This massive time resource management really keeps a game moving.
Power works. Give the players a ton of in game power that they can only keep if they invest in the game play or 'lost' power that they have to win back.How to get them invested is the big puzzle to solve.
I add tons of stuff to a game, like items and spells and secret devises. The sort of things the players need to find. Some is just simple stuff, some is game breaking and some is beyond amazing. Dor's hammer makes a clone of it's welder...just watch the players fall in love with having two characters. All sorts of divination spells that can give the caster REAL information,,lots of players LOVE these: watch a wizard player do anything to get a copy of Rays Replay of the Past (turn back time and see what happen in a place). The type of spell that gives a player a lot of in game power beyond just a pew pew combat spell.