I've been training my upper body using only body-weight for a long time now, and I recently started using a weighted backpack with nice tight straps (~+30% of my body-weight) directly over my center of mass. Then a personal trainer told me, it is much healthier to do Bench Presses instead, because they are an "Open Kinetic Chain Exercise" and thus put less stress on the joints. So two things: 1. Is he right? 2. If not, Are there any other reasons I would want to do bench presses instead?
Can weighted bench presses replace bench press?
Sure, they will work your muscles slightly different and incorporate core strength, which can be a bonus.
Is he right?
It uses more joints to do push-ups, because your vertebrae, knees, and ankles get involved, which might be bad in the long run if not taken into consideration, but again, it does incorporate core muscles, which is a bonus.
If you do the push-ups at home, he may be interested in keeping you at the gym in sessions with him. After all, he does make money off of it.
Things to consider..
bench press will work muscles differently, and it will be a more efficient way of achieving large muscle mass
keep your core and leg strength up to accommodate the extra weight that you add to your vest. The extra leg, lower & mid back, and abdominal muscles will support your joints and help prevent injury.
adjusting arm placement will use different muscles more and may help mimic some bench press variations. Try wide-grip push-ups (pectoral focus), close-grip push-ups (tricep focus), reverse-grip pushups, decline/incline pushups , etc.
bench press allows you to go heavier with a lot less spine, leg, and foot involvement
Your personal trainer is right. Your wrists will be at risk too if you do too many or too heavy pushups. So be careful. Read the story from this answer https://fitness.stackexchange.com/a/6489/25851, I also experienced the same wrist pain. The trick is to use knuckle on soft surface. But this solution is only for the wrists, not for all the joints mentioned in Craig's answer.
You don't have to give up pushups if you do bench presses. You should do both. They have their own advantages. Some people (like me) find their pectoral muscles work more with pushups. Also try to do your pushups after your bench presses and other pectoral exercises. Then, you need not to go heavy.