Brushed motor is likely OK (unless of crappy quality), the problem is that you don't have a controller. When suggested a brushless motor, the key is that a brushless motor can't run without a controller at all, so the controller is a no-brainer. A brushed motor can run without one, but it's very crappy.
A motor controller is a device which actively controls the motor current. By limiting the current to something the brushes and windings can handle, you can run against the current limit and even against physical stall for indefinite time, no problem whatsoever: it will run in constant torque mode. It's not an inrush limiter only; needing to regulate the current can happen, and often happens, in other situations as well, not only at initial startup.
A brushed DC motor controller is simpler than a brushless controller, since the only thing it needs to do is to PWM a bridge, using current measurement (and possibly a speed signal) as a feedback. A brushless controller does the same, but in addition, needs to handle commutation.
So look at DC motor controllers, with adjustable current limit. Off-the-shelf, they are pretty expensive, more than a simple series resistor, but with a proper controller, you reduce the system losses significantly, and it will last the lifetime.