Can wrist strap be attached to running PC

Don't get hung up about connecting things 'to ground', that is, that brown stuff outside the door.

What is important for you, your worktop, your tools, your components, is that they are all at the same voltage, before anything touches anything else.

That's all you have to achieve, it doesn't matter how you achieve it. People who say it must be done this way, or that way, are probably missing what we are trying to achieve. BUT, unless you get into a routine, and have a setup that allows you to do it effortlessly, it's real easy to screw up and get it wrong. Then, the best that can happen is you blow a component. Among the worst things that can happen are you half damage a component without realising it, which screws you up later (perhaps in front of customers), or you hurt yourself, which is why the safety resistors.

I see you are using a soldering iron. It is critical that your desk frame and conductive mat are connected to the soldering iron protective earth conductor for ESD safety, via a one megohm resistor for personal safety.

The safest most convenient common connection to use is the protective earth of your mains power supply. I have a three pin plug with only a series resistor'd ground lead coming out for the purpose. Everything else is wired to that. If you are not using any mains powered stuff, no soldering iron, no scope, no desk lamp, then any common connection will do, many people use the conductive benchtop, or its frame. You can use a radiator, but it doesn't add anything. Just establish one common connection, call it ground, and connect everything to that. Don't bus connections one from another, it's too easy to accidentally disconnect something you didn't intend, use a star connection of everything to your designated ground.

You could ground yourself before every time you touch something. But it's far easier to wear a wrist strap to ground. As this exposes you to an increased electric shock hazard, the wrist strap must be isolated from ground by a suitable resistor. One megohm is high enough for shock protection, and low enough for ESD protection.

Have a conductive work top, connected to ground. Metal is good, but hard on the eye, and you can't string circuit boards out across it. Conductive plastic is good, but expensive. Synthetic plastic furniture is a no no. Plain unwaxed unvarnished wood is often OK, it depends on the species and the moisture content how conductive it tends to be. Keep all your tools on the bench, and they will be at ground potential when you pick them up.

How do you take the component out of the bag? I have wept when I've seen even experienced engineers forget what the goal is, and juggle with a black plastic bag of components, trying to delay the point at which they touch them, while walking from component cupboard to their bench. They think they should do something, and have forgotten what to do in this circumstance. No. They should be achieving a goal. Everything at the same potential! Then figure out what to do to get that.

So sit down at your bench, put your wrist strap on. Touch the bag, or put it on the bench. Now you, bag and bench are at the same zero voltage. Now open the bag, take components out, and put them on the bench. All still at the same potential. There are other ways to do it, but what I've described is easy to follow, and worth getting into the habit of.

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.

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.