Or in other words, is the takeoff clearance ever given (implicitly or explicitly) before the aircraft reaches the hold position, so that the pilot can keep rolling through the hold line and to the takeoff roll?
No, there is no mandatory stop. Typical clearances given before departure are:
BigAir 123, hold short of runway 36.
BigAir 123, line up and wait runway 36.
Former phraseology in the US was "position and hold".
BigAir 123, behind landing A320, line up and wait runway 36, behind.
Note the word "behind" being repeated at the end of the clearance.
The interesting ones:
BigAir 123, are you ready for immediate departure?
It will be asked by the tower controller to assess if they can squeeze BigAir 123's departure between two landings, for example.
BigAir 123, cleared for takeoff runway 36.
It can be given to an aircraft still on the taxiway. In such case, BigAir 123, who will have proceeded through checklists and briefings while taxiing, will turn into the runway and takeoff, without having to stop at any point.
It is possible that the crew actually needs to stop, to finish their preparations. In such case, they will normally report not ready for departure, if asked. The tower controller doesn't want aircraft making unexpected stops on the runway, especially in a single-runway airport, this can result in the following landing having to go-around.
Also note that the word "takeoff" is used only in the context of an actual takeoff clearance or its cancellation. Otherwise the word "departure" is used, e.g., are you ready for departure? Also see the Tenerife disaster.
No, it is not always mandatory to stop at the hold line. Yes, the Tower controller can give you clearance to enter the runway (cross the hold line) before you actually stop at the hold line, in which case you're not required to stop there.
If the controller gives an aircraft the clearance:
Runway XX, Clear for Takeoff
Then the aircraft may both cross the hold line to enter runway XX and proceed to take off on it. Whether the aircraft stops after lining up at the end of the runway is up to the discretion of the Captain. Some pilots will chose to come to a stop after lining up and then select takeoff power, while others will just keep rolling and select takeoff power once they're aligned with the runway.
The controller may also add "No Delay" to the clearance. In that case, the pilot should not stop if it can be avoided or should at least keep the stop to an absolute minimum. If the pilots are unable to do that, they should reject the clearance and not enter the runway.
If the controller gives an aircraft the clearance:
Runway XX, Line Up and Wait
Then the aircraft is clear to cross the hold line to enter runway XX, but, after lining up, should stop and wait for takeoff clearance before proceeding to initiate the takeoff run.
Either of these clearances can be given while the aircraft is still rolling on the taxiway and, in either case, the aircraft may cross the runway hold line to enter the runway without stopping.
Two main instructions give a pilot clearance to cross a hold line: instructions to cross, and instructions for takeoff/line up. Neither clearance requires the aircraft to actually stop at the hold line. If the tower controller feels that clearance can be given early, they may do that. This is particularly true if a runway is not active and would have no conflicts for crossing aircraft, or if there is no other traffic for the runway at the time.
While the above answers are technically correct--you can be given takeoff clearance while taxiing--this is rare. The standard sequence is to receive a taxi clearance to the runway, where it is then mandatory to stop before the hold line. Once there, you get your takeoff clearance, allowing you to enter the runway.
You COULD request takeoff clearance as you are approaching the runway, obviating the need to come to a stop, but a towered airport is usually busy enough that the controller will want to see you at the line, ready to go, before giving you the runway.
Disclaimer: this comes from the perspective of a piston-airplane pilot. Because we have to do a run-up before takeoff, it is particularly true that we won't get clearance until we are ready at the line. A jet, which does not have this requirement, might get an earlier clearance.
Southwest 3828, Midway tower, runway 31C, line up and wait, don't plan on stopping.
That's in the audio but not in the subtitles, listen for it at 0:59.
ATC has a lot in motion, with Delta lining up on the cross runway and a short final approaching the other cross runway, and IIRC another flight on a 10-mile final on Southwest's runway. Southwest has gotta go pretty much right now or he'll lose the window of opportunity, and so ATC is clearing him past every stop point before he can reach it.
ATC also cleared Delta to roll past the hold-short line, but did not clear him to takeoff. If Delta had been paying more attention to chatter, he'd have the situational awareness to know what was going on.
Over in the railroad world, throwing clearances ahead of the train is SOP because of the high cost of restarting a train. I once saw a beautifully written train order that is the old-school version of this same thing, giving statutory wait instructions you won't be using:
Extra 4419 East [a lowly switch job] move Echo to Alpha, take siding for Train 1. Train 1 [the late Presidential Limited] hold mainline and wait at Alpha until 7:00pm Bravo until 7:10pm Charley until 7:20pm Delta until 7:30pm for Extra 4419 East.
This schedule is impossible for Train 1, it can't get there at those times so it won't wait. The order is for the benefit of X4419, who can now choose its meeting point with #1, without any further communication needed.