To add to fgysin's answer, if your campaign takes place over longer scales, with travel taking e.g. weeks, quad paper becomes useful:
In this case you'd use either a separate sheet, the other side of the paper or the right half for notes. Cross over days that go past with nothing significant happening, but use a number for notes if something is important:
Depending on the size of the paper and grid, you can fit a year on 2-4 pages, meaning even decades-long campaigns stay (relatively) organized.
Regarding who maintains it, as GM you can more easily set yourself reminders in the future or notes about NPCs if you do it. However, if a player does, you can still ask them to mark down e.g. a star without telling what it means. Sneaky way to hint at a deadline...
You said:
I think the core of the problem is that we are playing out every awake moment of the characters - there are no visible scenes, only continuous flow of character actions and world reactions.
I had a campaign very much like that a while back. It took them multiple sessions to just do one task walking around an island. They didn't mind at first because just the experience of camping in-character was entertaining, but we did eventually agree to settle into a breezier pace.
In my experience, the key is to think about movie pacing. Do they show every waking moment? No, they say "Let's go to a place" and then the next scene is "Ok, here we are at the place, let's do this!" You can absolutely do the same thing: "OK, about 4 hours [or even 4 days] later, you arrive. As you approach, you hear..." Basically, get to the good stuff.
Note that this approach requires trust from both parties. You have to trust that on the one hand, your players will interrupt you if they wanted to do something particular; and on the other, that their characters are reasonably competent and act like it. Your players have to trust that you won't pull the old "You didn't say you got dressed this morning so now you're all naked and freezing" - if the adventure doesn't specifically call for resource management as part of the tension, and players want to clarify things a little bit retroactively, let them.
PC: "OK, I offer the princess a drink."
GM: "A drink of what exactly?"
PC: "Of water from my waterskin, which, it goes without saying, I filled
up at the well before we left town."
GM: *the briefest of pauses to consider if their character is like that* "...Right, naturally. She accepts gratefully."
If you and your players don't want to spend time listening to each other fill waterskins, you have to actually let that go without saying. That doesn't mean you can't show more mundane parts of life, but usually you'll do a bit of that towards the beginning as people get to know each other and establish routines, then count it as done and gloss over it the next time. For example, you can ask them how they do watches once if that's applicable, let them argue it out (preferably, for my taste, in-character), and then assume that they do them the same way in the future.
In my case, part of the problem was trying too hard to avoid metagaming - so I'd roll lots of random encounters including harmless animals, but the players would treat each one seriously and look around for danger, which took up quite a bit of time. It's okay to assume that's going on as you're traveling, and only actually ask your players to roll to see if they notice any danger when there's actual danger to notice. Along the same lines, you can ask in very broad terms if they're going through a dungeon carefully searching for traps/treasure vs. moving quickly (the DMG has specific guidelines for this), and then just roll for them, use passive Perception, or ask them to roll quickly when there's actually something to find. This saves your players from describing, in detail, how they search each room... which can be part of a good game that tests player skill (you don't find the gem in the dresser unless you actually tell me you're searching the dresser) if you're all into it, but definitely slows down the action.
In general, just remember that you don't have to roleplay everything out just because it happens. For skill rolls, the Angry GM (warning: he is indeed angry and a bit vulgar) advises to only roll if there's a chance of success, a chance of failure, and a consequence for failure. You could almost say the same thing about roleplaying - only make the characters go through a scene if there's some decision to be made, some opportunity for the characters to really show who they are, and/or some way they could really screw things up and then save the day. That's what good stories are made of, right?
Best Answer
Time limits the players don't know about are useless.
You don't know where your players are gonna go, or what they're gonna do. If you have events happen off screen that they can't foresee, there's no real tension, and they might decide to go somewhere else. You can have timed events for your own fun, but roleplay is about collaboration and interaction.
You should adjust your events to be more active and interactive.
As I understand it, they left the macguffin on a random truck, and the truck driver is gonna drive off a bridge and die? Did you tell them that this truck would likely drive off? They left no one there who could tell them? Regardless, the macguffin randomly being lost isn't that interesting. Instead, have things happen with it.
Blades in the dark has a fun mechanic, of clocks. You make a clockface, divide it into several fractions, and each time a big event happens you colour in a face. So, for this truck-
The truck driver finds someone invaded their truck, and goes to analyze the macguffin. The next time the PCs go to check on the macguffin, it's lost.
The truck driver arranges an auction for the macguffin. The PC's contacts can tell them it's being sold, as everyone knows.
The big bad goes to buy the macguffin. They can see some assets being sold off to pay for this, or they can see troops being moved to ambush the auctionee.
The big bad activates the macguffin. Whatever the macguffin does happens.
At any point in this the PCs can intervene. If they don't whatever happens happens.
Having the macguffin randomly fall in a hole is a lot less interesting than a giant auction of evil.
This serves as general advice when you have a time limit. Don't make it quick and hard to see. Make it extended, and escalating in it's badness.
An example, just as you asked, of how you might handle a cursed tomb is The Mummy.
The Mummy has these steps. Step 1. The curator burns the map to the tomb, because people who go there never return. Step 2. A knowledge check by the protagonist says the tomb is a portal to hell, and they see Americans going to the tomb. Step 3. Ardeth Bay shoots at them and says "Leave this place or die." Step 4. On the way into the tomb, supernatural monsters consume some of the American's natives. Step 5. On the tomb of the Mummy is a curse, saying that whoever opens it will be consumed to resurrect the Mummy. Step 6. Everlyn decides to read from the book of the dead, and is warned that doing so is dangerous. Step 7. Everlyn reads from the book, and resurrects the Mummy. Step 8. The mummy eats the Americans to regain it's strength as the curse said it would.
Every step on the way is well telegraphed, and they had many many chances to avoid this danger. That is ideal.
Make time limits interesting.
That's the key aspect. They should be slow, well telegraphed, and give the PCs a chance to shit their pants as things get worse. That is what TRPGs are about.