Another thought. Apparitions seem drawn to/shaped by people's fear of them.
I can see it being easier to train them not to attack people who aren't terrified of them, which is a reason not to choose a shape that regular humans will be very scared of.
It's not the only reason to do it—there's the inherent danger and deadliness of big cats, the monsterfication of the mount, disguising its origins as a red rider, and also a subtle nod to the Gold Order's Dark Hounds by having a Dark Feline.
... Although I will admit the Ranald connection is still a rather prominent reason.
Thread narrative seems to influence Mathilde's thought process, and cats seem to have been all about Ranald.
Big cats probably aren't a very culturally significant thing to her either. She didn't watch nature documentaries on TV as a child or see books with illustrations of them when she was in her formative years.
Ultimately I'm less optimistic about the performance of this thing than you are, it seems. The fact that it has a mind means we can probably make it understand what we want but ultimately while it's intelligent, it's an alien intelligence and I think you're assuming a bit too much about how it's going to get to that one person in the crowd (I think it's natural tendency is just to go through them).
The thing is, currently it's perfectly happy to leave people who aren't its favoured targets unharmed. I don't see why training it to change its favourite targets would change that attitude.
If we had to teach it not to attack civilians that would be one thing, but that seems like its default behaviour.
I think Red Riders are imperceptible save by wizards, but I don't think they're incorporeal. I don't think they can selectively phase through people or objects. They're horror movie monsters that would batter down the door, not walk through it, I feel. Rotwyrms are specifically noted to be incorporeal, which strongly suggests the others aren't.
This is supported by Boney saying that a monster without magical weapons could push away an apparition, but being unable to harm it would eventually have its limbs disabled by its attacks. That suggests they're by default invisible, not ethereal. The Ulgu coating would make them visible to everyone when manifest, but I wouldn't have thought it could grant incorporeality.