I think the majority of Soulcasting is done using fabrials, there might be a few special cases besides Jasnah and Shallan though.
Can't tell you exactly without knowing more about soulcasters, but it is likely that the spren imprisoned in the soulcaster is gone, released (or destroyed?) when the soulcaster was broken.
That is, of course, if soulcasters follow the examples of more modern fabrials in relying on imprisoned spren.
According to Harakeke's translation (thank you) of Navani's notebook:
The cut and type of the gem determines what kind of spren are attracted to it and can be imprisoned in it. There must be thousands of possible combinations.
Once a spren is captured and the gem infused with Stormlight the fabrial can be used in machines.
It's the
gems that attract and imprison spren, and by infusing the gem with stormlight the gem becomes usable. The fabrials themselves (Soulcasters and Shardplate included?) aren't what's imprisoning the spren. I wonder if this is related to how cut gems store more stormlight than uncut.
Because it hasn't been put back together "perfectly", just well enough that visual and tactile inspection can't tell the difference. It is an ancient magical device, and mere physical reassembly is not sufficient to repair damage to the magical portion of its workings.
This is what I'm going with.
Also: The Soulcaster was found in an inside coat pocket, sheared through a chain and a gem setting (I believe there are 3 gem settings and 2 chains that connect to 2 rings, quite a bit like the angreal that Nynaeve used in WoT, which is a possible influence on its design). They had a jeweler repair it to the degree that Shallan can't visibly tell that it was broken.