First off, you are wrong if you think that bodyweight exercises cannot injure you. Even something as simple as doing pushups with bad form for too long can have a detrimental effect on your posture and cause back issues.

Also, that's a false comparison. Nobody said that you need to get off the couch and go straight to squat 500lbs to be in shape. You need a proper training plan with reasonable progression, and you need to start as low and as light as necessary, no matter the training stimulus (BW, barbells, dumbbell, kettlebells, bands, ...). The difference is that weights can provide an external force that can be scaled to your strength in many ways that BW training cannot, because in BW training the only weight is your body and you need to use leverages to modify your training stimulus.

Finally, BW versus weights is a dumb debate. They complement each other, and a well-rounded athlete should do both.