I had a bad back for many years. I could have left work on disability due to a lower back injury.... but I didn't. The sedentary lifestyle I settled into while trying to avoid repeated back injuries left me fat and with foot problems by the time I was old enough to retire.
I bought a sit-up-and-beg style used bicycle just to get some fresh air. I loved cycling again... and was soon given a road bike by a friend. The road [style] bike distributes my weight across my contact points... with very little pressure on my lower back.
For me..... road cycling changed my life.
I no longer suffer back problems... to speak of. My health is greatly improved and I feel decades younger than I did a few years ago. Cycling is not a cure-all by any means.... and it does introduce new health risks as well. But I'd take a doctors advice over "people" on the internet any day. If the doctor didn't mention a brace I wouldn't worry about it.