DC Motors with a feedback line (so called PID Controller; https://en.wikipedia.org/wiki/PID_controller) would be the best way to do it. You need the motor as an actor and some sensors that sense the direction of the axis's of the gimbal. The PID controller should be very fast for smooth operation. (Perhaps an ESP32 is a better MCU for this purpose).
The PID controller uses a motion sensors.
It's good to have strong motors that do not react "nervous" on on voltage changes. You probably will have to use motors with gears. Even fast PID controller won't be able to keep your camera from turning cartwheels if a 10.000 cycles per minute racing motor gets full voltage applied.
I searched the internet for videos of such a setup but the only thing I found, are the commercial (?) gimbal "alexmos" "how to" videos. I'm with you that it is hard to build such a gimbal without much knowledge of how PIDs work. I'm also not sure if there is an arduino that's fast enough to realize a PID that gets its feedback from motion sensors and drives dc motors with this information, then get's feedbach of the motor movements (from the motion sensors) and so on.
And probably a better/stronger servo and and very fast PID setup might eventually smooth the gimbals movements.
Here is a link that might help you as a start point for PID (if you are not discouraged yet):
And even professional setups can make problems: (It's a matter of the PID parameters)