Both gyms I have attended, they don't exactly throw you to the wolves, but you are live rolling the last half of the class on your first day, and if you are morbidly out of shape, it is not fun. I've tapped before due to just having someone on top of me. Couldn't breath, people asking "are you alright, are you okay, are you gonna die?" They look at you funny when you have to tap and get some air, like they don't know what they just did to win.
I also do not agree that nothing prepares you for bjj. Any cardio prepares your lungs and cardiovascular system to supply increased oxygen. Professional bjj athletes do strength and conditioning outside of just bjj, and they don't do that because it's ineffective and doesn't prepare them for bjj. Yoga, pilates, and 45 minutes on an elliptical per day will definitely see an easier transition into bjj than just diving in, and sometimes that extreme difficulty plus the embarrassment of being "the fat dude" is enough to make people quit quickly as well, as opposed to never starting.
Beyond that, bjj is an intense physical activity, and sudden exertion by someone who has been sedentary could cause heart palpitations or cardiac arrest. It definitely is good to slowly ramp up to being able to push your heart rate up there. It's bad for someone with like 90-100 resting heart rate that never exert themselves, to suddenly be hitting 200 in a short burst(which is what happens to someone who is badly out of shape). I would say anyone over the age of 30 that has been extremely sedentary honestly NEEDS to go to the gym and do some work on an elliptical or bike, before they roll full speed with another student. Maybe light rolling with an experienced student or instructor, drilling, technique, but rolling will have your heart pounding in a hurry, and that could be bad for someone that may already have some heart problems brewing or diagnosed.
I do understand how most people use this obstacle to never train, much like a "i'll start my diet next week," but it is not easy or even safe for someone to immediately go from their divot on the couch into rolling. Even when I hit the gym hard for 2 or 3 weeks before going to bjj class, it's still not easy.
I think the best response to "i need to get into shape before bjj" is "you can go at whatever pace is good for you, just tell your instructor and partners what your limits are. If you have any questions about your heart's condition and have been very inactive for an extended period of time, see a doctor first and begin with some very light cardio and adjust your diet to be healthy. Size yourself and order a gi as a motivator, and attend class even if just to watch."