There are several considerations for your wrist strap.
A) Safety for you
B) Safety for your fragile components and kit
C) Process, use them correctly
D) Grounded?
A) Your safety
Your wrist strap should include a large amount of current limiting before connecting to a real earth. In practice, this is a 1meg resistor in the strap, and/or where you clip it to, and/or a connector that you plug into an earthed outlet to contact the ground pin.
B) Your kit
There is no point grounding yourself, if the stuff you're working on can float to any potential. Use a conductive sheet, and place all your tools, components, work in progress, on it, and ground the sheet (via a safety resistor) as well. Metal foil, sheet or a tray will do. In industry they tend to use conductive plastic, which is nicer to work on.
C) Process
Having the right equipment doesn't help if you don't use it correctly. When your PI arrived through the post, it was (obviously) not connected to your grounded sheet. At some point, you have to connect them, and it's at that point a damaging charge transfer could occur.
When you connect them, make sure the point that connects first is a grounded point of the PI, a connector shell for instance. Before you unwrap a component from its conductive bag, or pull it from the conductive foam, touch the bag or foam to bring it to the same potential as you (ground).
D) Grounding?
Once you, you tools, your components, your work in progress are all at the same potential, it doesn't matter whether the whole equipotential group is actually connected to ground or not. In practice, it's a whole lot easier to keep track if it is, and as soon as you use a grounded soldering iron, 'scope or power supply you have that earth connection, so you may as well start off with it.
The point about static electricity is that there's usually not that much energy, i.e. charge, building up over a short time.
Therefore, a couple kOhms resistance don't make that much difference - your body voltage would break down over that.
I'd argue that the fabric ones are better for most usage scenarios: they are simply more comfortable to wear, and that makes a difference in usage, and also, humans become sloppy over time – and that'll lead to you being more likely to take off the wristband "earlier".
Is there any change of static build up on the insulating outside of a stainless steel anti-static wrist band?
I'd say: no. Not at all. Ok, it depends on the conductivity and thickness of that lacquer, but assume the following. I'll try to draw a worst case scenario.
You take your angora bunny to your workplace.
For some reason, that bunny gets aggressive/bitey when you don't pet with your wrist, so you pet it with your wristband, and by doing so, you're constantly stroking with the outside of your wristband.
Charge builds up on the wristband's outside. However, that charge is either very small, or directly discharges into the metal part.
So you're fine.
Someone notices you and your pet rabbit at work and fires you, but not without offering the position of chief rabbit safety overthinker in his night-time job, organizing rabbit fairs. You become famous and rich.
Not that bad for a worst-case scenario.
Best Answer
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.
Excerpt:
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):
(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.