This is one of those things that works fine... until it doesn't.
I think LaBella is the only string manufacturer that specifically warns against this but I've seen flats from other manufacturers unravel at the saddles too, when strung through the body. It's not something that will necessarily happen the first week or month, or even the first year. But based on what I've see at the workshop it's looks very much like going through-body can shorten the life of flats. At least SOME types of flats.
Personally I would only do this with flats that are specifically designed for it (like LaBella's Mustang flats) or MAYBE with certain types of extremely flexible, lighter gauge flats.
It's only a set of strings of course, which can be replaced relatively cheap. But with flats you want them to last. Preferably a very long time! Most flats need to mature over YEARS to unleash their full potential. I would be very angry with myself if a nicely broken in set of flats unraveled at the saddles, because I had ignored all warnings and strung them through the body...
(...which in my opinion offers no tonal benefits whatsoever as long as the break angle over the saddles is also sufficient when strung through the bridge, which is nearly always the case, except on weird bridges like on Danos, which don't offer string through anyway, which means you've got no choice in the matter...

).
To sum it up:
If I break a set of roundwounds I get mildly annoyed.
If I break a set of broken in flats...



Why risk it?
(Unless your bass is through body ONLY and you REALLY need to install flats on it and you are fully aware of the potential long-term risk.)