Edition-agnostic question here, and answers based on different editions might be of interest.
First, the background:
Some years ago in my current campaign one of the PCs - a Human Fighter - suffered a mishap where pulling a card from a modified Deck of Many Things stripped away his soul but otherwise left him hale and hearty. He ran for a few in-game years like this (and, somewhat incredibly for both the player and the game, stayed alive the whole time) while slowly coming to realize his odd condition, then eventually - with some very high-powered help - he got a soul stuffed into him that nobody else was using (it was extracted from the soul gem found in Ghost Tower of Inverness).
The ramifications of this were both good and bad. The good: he became completely invisible to most undead, who see only the soul or spirit of the living. He also became immune to a few (quite rare) effects that specifically target the spirit. The bad: if he died he was done. No revival in any manner, no speak with dead, nothing.
And so, my question to you all:
Would you allow a PC to lose its soul or spirit like this and yet still remain alive and playable?
If no, why?
If yes, what would you have as the ramifications of having no soul?
Personally my take would be similar to what you're describing.
I don't really like takes which involve people losing empathy or humanity if they lose their souls, because, it's very clear in our own scientific world that, if there are souls, they're not needed to explain empathy, humanity, decent or the like. Personality seems to mostly a fusion of memories/experiences, how well the brain is functioning, and the endocrine system. You don't need to force/jam in a soul to try and explain stuff that's already well-explained.
Thus someone who lost their soul shouldn't change in personality or the like from the mere loss - if they didn't know they'd lost it they shouldn't change at all. If they did change it would be likely from the trauma of finding out, and perhaps subconsciously acting into stereotypes of how they thought a "soulless" person would behave.
I see the soul, unless specified otherwise by the setting, as more of a "restore point" or "archive" of a person's mind and body - this works pretty well with most fantasy cosmology and magic systems.
This was my rationale for letting the soul-less character survive: his physical mind - i.e. his brain - kept his memories and personality going to the point where an observer couldn't tell the difference between his previous and new versions.
Yeah and his endocrine system - that's a big part of who we are too (something often overlooked in sci-fi stuff about body-hopping and the like). But I agree. Unless you jam the soul in and have it start replacing the role of parts of the brain and endocrine system, there's no reason someone without a soul should be "passionless" or "having no empathy" or "no remorse" or the like.
I'd also say I personally object to that, because the same negative things people often like to assign to characters without a soul are also assigned to atheists (and sometimes even agnostics!) by various religious leaders and politicians (and their followers) in the real world, and can have real negative consequences for how people who are atheists or the like think and feel about themselves. We can't discuss religion/politics in detail, but I wanted to bring this up primarily as a diversity issue - I don't think it's great to ape anti-atheist stereotyping as "what it's like when you have no soul". Obviously you're doing the opposite of that, thankfully!