Indeed, the red rose is a rather docile bird-catching spider. It hardly attacks your hand, but even if it doesn't attack you, the hair on the adult or sub-adult will make your hand itch (it can be washed with soap and hand sanitizer). I suggest you use it as little as possible. Perhaps the reason for keeping bird spiders is not that they can crawl around in your hands, but to observe the pleasure of their comfortable life in your breeding box.
If you get bitten, you don't have to go to the hospital at all. Just disinfect the wound or wash it with soap/hand sanitizer several times. If you really can't let it go, you can go to the hospital (the doctor should just disinfect your wound).
In short, as long as you start with less, all problems will "die".