First, the mat (and the connection to mains earth!!) should have a resistance that is high enough to cause no (extra) harm to you when 'normal' Voltages (let's say up to mains level) are involved. Think Mega-Ohms, and preferably more than one resistor in series to counter single-point-failure issues.
As for the open/closed loop part: think of your body as one plate of a capacitor, the other plate being the earth (in the form of all earthed metal in your surroundings). Now when you touch your mat, there is a closed path. The part that you think of as open (you and the earth) is a capacitor.
Another way to look at it is that for static electricity (as opposed to the more-or-less continuously current carrying electronics we are used to) a closed path is not a requirement. An excess of charge tends to spread itself out and preferably find a place with a shortage of charge to combine with. This is a temporary phenomenon, which lasts until the charges are evened out.
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.
Best Answer
A proper ground strap connection should show 1 to 10 megohms between the part that contacts your skin and the ground terminal of the power cord. This resistance is there to prevent electrocution through the strap.
The point of a ground strap is to dissipate static charge. A large resistor does this slowly enough to not "zap" you, but also doesn't turn you into a large ground rail yourself.