What type of illness exactly would you be concerned about?
Horses on a particular farm are typically turned out in herds/groups/pairs with shared water and hay sources, often mutually groom each other with their mouths, and eat the same grass, which is contaminated with at least some level of each other’s manure/parasites. Even horses on private turnout rarely have a paddock to themselves–often other horses are turned out in the same paddock at different times. Horses on a particular farm are also handled, fed, watered and groomed by the same person, and grooms do not typically change clothes/wash hands/use a foot bath between horses. Grooming tools are often shared between horses (unless owners are doing their own grooming). Horses use the same crossties and wash stall, heads are often hanging out in the aisle, and many barn set ups have windows or bars between stalls so that horses can communicate, and many paddock setups allow horses to visit over a fence. Horses at a training barn often share bits and bridles without a second thought. Often grooms will take buckets and feed tubs out of stalls to rinse them and they often go back in a different stall to a different horse.
Realistically, I can’t think of an illness that a horse would be exposed to by sharing a stall that they wouldn’t already be exposed to via simply living in the same barn.
Mind you, the situation is completely different if stalls are shared by horses coming in from other barns, such as for lessons or clinics–I would absolutely view that as a biosecurity concern.
I have no problem with people wanting their horse to have it’s own stall, but that is a human desire, not something that has any impact on welfare of the horse, unless the stall is not being cleaned or bedded adequately, and that is a separate issue entirely. The OP discusses that her horse smells like urine, however, large warmbloods can produce copious amounts of urine and manure and some horses specifically prefer to urinate in the stall. Even with a spotlessly clean and well bedded stall, it is possible for a horse to come out of the stall in the morning smelling of urine. Bedding costs are on the rise, so I recommend that if the OP does change barns that she specifically take a look at stalls and look at how much bedding is used and what type.