Since most weighlifting injuries are to the shoulders and back, I've decided to remove exercises that target these two major muscle groups. They get some limited involvement from other exercises, so they're not completely ignored. I had a minor shoulder injury years ago from lifting, and I know people who have really screwed up their back, even from abs workouts which put a lot of strain on the lower back. Suffering an injury, especially one that could become chronic, is just not worth any benefit from targeting those muscle groups. A personal trainer might say, "You just have to do them in proper form with proper weight". Yeah, just do it perfectly and you'll be fine. Who is perfect? I'm not a bodybuilder posing on stage in a speedo, so I don't need to be perfectly proportioned. I'm doing this for my health. So in addition to getting a safer workout, it's also a shorter workout, a double bonus.
Ignore these areas at your own risk. The trainer would be right. Just leave your ego at the door, go light and work on form.
I'm assuming you mean the lower back when you say "back" and not the mid to upper back region. I agree with you that those two areas are definitely common injury points, but I'm not sure about the concept of avoiding direct exercises entirely and relying solely on ancillary movements. By not directly stimulating those muscles, I would argue that you're weakening those muscle groups over time and setting yourself up to get injured even more easily. What I do is just lighten up the weight dramatically and do lots and lots of reps. For example, stiff-legged deadlifts will definitely work your lower back a lot, but instead of doing that movement with a barbell with a plate on each end. Just grab a pair of 20 or 25lb dumbbells and do 5 sets of 20 reps. By the time you get to the 100th rep, your lower back will be pumped but you won't ever feel like you're going to pull or snap something like you would doing 5 sets of 5 with heavy weight.
The only shoulder isolation (exercises) are iterations of overhead presses. The back workout is the only essential workout. Tread lightly with an injured back, but if you cannot do back then stick to cardio for now.
I would second the advice as given by @Baron Instead of ignoring some muscles (or muscle groups) entirely I would use a weight which is low enough that I can perform sets of 15 to 20 reps. And do 3 to 5 sets. In this way you do train the muscles but stay far away from risk of injury due to overload.
Yeah, do some cleans with low-weight on an oly bar. Like a pair of 25s for 90 + collars. That's @CaptainObvious' one-rep max.
For your shoulders, you could try a pike press. However, beware of videos on the Internet showing you how to do it the wrong way. You should try to emulate the movement of a regular shoulder press, keeping your elbows in and flaring them no more than 45 degrees away from the body (unlike the videos that show a 90-degree flare). Also, since the weight in a standard shoulder press should be in front of you, be sure to lower your head to the front of your hands rather than at the sides which causes the elbow flare. Depending how you position yourself, you can make the movement easier or harder, and you can let your shoulders tell you how much range of motion they will allow. Remember: muscle pain is good; joint pain, not so much. As your shoulder gets stronger, you can lower yourself so your face, or even chin, touches the ground (towel) rather than just the top of your head. A slow and deliberate speed during both the up and down movement can also heighten the resistance as your shoulder improves. For your back, if your issue is your lower back then you can do inverted rows, either with a supinated or pronated grip. Slow, deliberate form is key since it is only a body weight movement and you want to make it challenging. As you get stronger and the movement gets easier, you can elevate your feet. Apart from maintaining a "plank" position throughout the exercise, your lower back is out of the mix. Anyway, good luck with whatever you decide.