Smart switches are not equal partners like old style 3-way switches. They are "master" and "remote".
This model of smart switch is a "master" and requires certain models of "remote" which require "neutral" (white wire) to be present in both switch boxes. That is not always so, as the electrical code only recently started requiring it.
So I'd first check the other box - if there's no neutral in it, you cannot wire a remote (though wireless options may exist). Useless old work can be dangerous, so tidy up by removing the 3-way switch, wire-nut the common to one of the messengers (trial and error will reveal which), cap the other one off, and put a blank wall plate over the hole.
If a white wire is already wired to a terminal on the old 3-way switch, bad news - that is NOT a neutral, but one of the messengers (the two inter-switch wires). Before the code change, they often used common 14/2 Romex for messenger, and one of the wires is white. They're supposed to paint or tape it.
If there is a neutral in both boxes, you can wire a remote. First, figure out which box is upstream (closest to the service panel aka breaker box). It will contain a wire that is always hot - regardless of the position of any switch.
Then review the diagram (which you already linked) showing you the wiring scheme for whichever remote you bought. You will need to re-use both of the old messenger wires.
From your description it sounds like you have the intelligence to understand this stuff, but are simply lacking certain nuggets of information. You'll want to read up a lot on "switch loops" and particularly "3-way switches". It will all make sense pretty quickly, then.
Oh, and one more thing that's a bit harder to uncover: In America, wire colors do not have firm meanings. Green always means ground. White or gray always mean neutral, unless marked or in switch loops and messengers - the goal being to build the necessary circuits with readily available multi-wire cable even though it's the wrong color. "Hot" lines are everything else - including marked whites and (implied) switch-loop whites, and there is no official standard of how to designate certain types of hot. (however in commercial facilities, consistency is required.)
Best Answer
I would not worry about adding a “wire” to the switch either the box grounded those nice bright brass yokes being connected to the metal box will be more than enough grounding (in the US it is code compliant to use this same method with some devices). Your switches look to be in exceptional condition I hope the contacts are also if so they may last another 100 years.