The maiden example was just an example, but its a useful example. There does not need to be a clearly stated conditional time pressure on every mission. But DMs should always act in such a way as to allow their players to know that events happen in the world around them according to a logical, progressive time. I don't fixate on the minutes and hours of the world, but I do focus heavily on the days, beginning most adventures with a mention of the date and, after camping, reinforcing the fact that it is a new day. If you train your players right, they create their own pressure, because they are not sure what the ramifications of delay might be. Who knows how long they actually have until the big bad is summoned, the girl is killed, the treasure is spent, or the dragon summons help. If they go away and leave a situation uncontrolled, they can be fairly sure that things will likely have changed while they are gone as the other side continues to act. Or at least that is always the assumption on their part, even when I, the DM, am going to change up very little.
I guess it boils down to a play style where, while my game may revolve around the PCs, the world never does.
When I say there's not always a time pressure, I don't mean, "No, there totally is one, it just isn't stated up front." I mean that sometimes there really isn't a time pressure. A couple of actual play examples I've seen over the years may help illustrate this.
The PCs, being adventure hungry new guys in town, ask around about possible places of interest. They are told of the ruins of a monastery near town, where a valuable jewel may still rest, after all these years, but the townspeople are afraid to go near the ruins(Yes, this was AD&D1e. Yes we started the campaign copying the books' example of play. It was our first game). Giant spiders, ghouls, traps, that place was dangerous. Luckily, there was also nothing stopping us from leaving if it got too hot. The treasure wasn't going anywhere. The townspeople were afraid of the place, and we were the only adventurers in town.
Example two, a botched funeral ritual accidentally desecrates a graveyard, and the dead begin to rise. Our heroes, being do-gooder type chaps, decide to help. We clear the dead from above ground, and find that there are many more in the crypts, so we start to clear them. Spells/hp/etc run low, so we pull out, board the entrance, and set a watch while we patch up. The only undead there were corporeal, and there was only one entrance.
Now, in either of these cases, the Dm certainly could have created a pressure, ad-hoc, by adding in another group of adventurers after the jewel, for instance, or another exit to the crypts we didn't know about. Were I DMing(or rather, were the me of today DMing, example one was actually me DMing, so many years ago) I probably would have added just such an element, to keep the players on their toes. But does that not go against your stated idea of the world not revolving around the PCs?