You can get very close to what a good bike fitter would do on your own. I wrote a little bike fit primer a while back. It's here:
https://www.bikeforums.net/21296948-post3.html
Pedaling definitely works one's back. Back pain is one of the most common problems we hear about. The thing is, when you pedal you are activating what's called the "posterior chain," which goes all the way from your heels to your shoulder blades. If you watch someone pedaling in a tight jersey, you can see their lower back muscles flexing. There's not a lot of force there, but cycling is an endurance sport and over time it adds up to pain. The pain can be from two different things. First, there is muscle pain from the exhaustion of individual back muscles. But once those muscles get exhausted, they no longer support the spinal column like they should and then one can get real back pain.
While you're doing your bike fit, also look in the mirror at your back. You want to rotate your pelvis somewhat forward so as to straighten your lower back as much as is comfortable.
When people say "core," too many think "abs." I've never encountered a cyclist who complained about sore abs. It's always the back. IME the very best back exercise is bike riding. It takes time to get stronger, just keep at it and don't push it way into the pain zone or take ibuprofen for it. Stop and rest and stretch a bit. Sitting on the curb can feel really good. Probably the best weighted back exercise is to pick something up with a straight back. With knees straight or very slightly bent, never locked, pick something up and go all the way to upright back, keeping your back straight. There's always too much talk-talk in these videos, but this one is better than some: