The 1meg resistor is needed to safeguard the user from faults from other equipment connected to mains earth.
Keep in mind that the wrist strap is a permanent connection to the electric system of the building. If another piece of equipment experiences a fault, there could be a big fault current through the mains earth wiring system. That means that, in unfortunate circumstances, the mains earth terminal could reach a dangerous potential. In this case, the 1Meg resistor limits the current from the earth wire trough the user to a safe limit.
See this Wikipedia article about Earth Potential Rise, for example.
The resistance of the Earth is non-zero, so current injected into the earth at the grounding electrode produces a potential rise with respect to a distant reference point. The resulting potential rise can cause hazardous voltage, many hundreds of meters away from the actual fault location.
Therefore the earth wiring system (and your wrist), due to its low resistance, is roughly at the same potential of the point where the fault current enters ground, whereas your feet (several hundred meters away from that point) are at a lower potential. Without that 1Meg resistor: ZAPP!!!
EDIT (to address downvoting and clarify my answer)
Since my answer has attracted a couple of down-votes and some criticism in the comments (not necessarily related, at least not apparently) I feel compelled to clarify something, but I'd like also to remind downvoters what down votes are for: for answers that are not useful, not on topic or plainly wrong.
First: I was told that regulation doesn't require the 1Meg resistor for the reasons I stated. My answer: I never stated that my explanation was related to some regulation (I didn't even know there was a specific regulation for wrist bands - BTW, I'd like to see some reference), but I concede I could have been more explicit.
Second: As I wrote in a comment, I admit that my scenario is less likely than, for example, touching a live wire or an ESD event whose rapid discharge could cause issues. Nevertheless, as someone said in a comment, You only die once! Faults in electric systems do happen, and often they are not under your control, so no level of care from your side could prevent them, you can only (try to) prevent the consequences. Hence the scenario I depicted is, IMO, well worth considering (so it is on-topic and it is useful). Moreover, the question in the title is Should there really be 1 MΩ resistance between an anti-static wrist strap and a pc?, not something like Why regulations impose a resistor there? or What's the most likely scenario that resistor is put there for?.
To further make my point you can see this article on Wikipedia about Stray Voltages. Not everything is directly related to what I'm saying, but the part on Neutral return currents through the ground is. Excerpt (emphasis mine):
Stray voltage became a problem for the dairy industry some time after electric milking machines were introduced, and large numbers of animals were simultaneously in contact with metal objects grounded to the electric distribution system and the earth. Numerous studies document the causes, physiological effects, and prevention, of stray voltage in the farm environment. Today, stray voltage on farms is regulated by state governments and controlled by the design of equipotential planes in areas where livestock eat, drink or give milk. Commercially available neutral isolators also prevent elevated potentials on the utility system neutral from raising the voltage of farm neutral or ground wires.
(I didn't have the time to search for an article involving grounded humans instead of grounded cows, but you get the point.)
Bottom line: connecting a human body to any low-impedance path that could possibly rise in potential is dangerous and life threatening, so proper safety measures should be in place.
You are over-thinking this. If you want a handy ground reference to your laptop. you can clip onto the ground/shell of a USB or HDMI (etc.) cable plugged into the laptop. No need to sacrifice a perfectly good cable.
If you really want to make a dedicated cable, use some old USB cable or something easily replaceable.
In reality, as soon as you touch the screwdriver to the screws holding the laptop together, you have ipso-facto "grounded" yourself to the laptop.
The reason to ground the PC case has nothing to do with ESD: it's a measure against PC case going live. If the case is grounded, a live wire touching the case will create a prompt short circuit, which will (hopefully) trigger the circuit breaker and instantly remove the power.
Grounding yourself via wrist strap has nothing to do with your safety: it actually makes things worse for you. If you touch a live wire while not wearing the strap, the current you'll get will be limited by your body's capacitance and leakage via your shoes etc. Should that happen while you're wearing a grounded wrist strap (without a resistor), that current will only be limited by your body's resistance, which is not that big at high voltage. That's why you need a resistance in series with your strap.
Now, it is true that your PC case which is grounded without any resistance remains a shock hazard: if you manage to touch it with one hand, and a live wire with the other one, you'll get a bad shock. But this is quite improbable and requires you to do something very bad (touch a live wire) in the first place. A case going live, on the other hand, is not an impossible event: a simple bump on the side may deform it hard enough to touch something on the power supply PCB. Should that happen, you'll have that same very bad stuff right in front of you and without any warning: a 1MOhm resistor can withstand 230V indefinitely.
Edit: after the exchange in comments it became clear that you don't need protective Earth. In that case, I would go for Plan A, as it has more wiring behind the resistor and less wiring connected directly to Earth. Those resistors are installed for your safety as explained above, so it's clearly better to have more touch-safe wiring on your desk.