ROS (Industrial) can be an option for all the "higher level" stuff, and there are a lot of (open sourced) robots that work with ROS so that is one way to search a bit more direct.
Chris Anin has opensourced his AR4 (or is it AR5 now). It's a robot + controller, but as most small robotic arms, it's based on stepper motors, so the low level electronics hardware is different. But it is a current project that is being developed further (or at least, it was when I checked recently).
Any uC based BLDC motor driver can be used to drive a brushed motor, by just using two of the three half H-bridges, but the EUR 10 board you found is probably adequate. Many uC's have hardware onboard to track the position of a quadrature based encoder, and that makes software development easier, (else you would need an ISR for each of the encoder channels).
If you want to design / build your own motor controller (PCB's). There are a lot of IC's that implement a full power stage of an H-bridge. These things are used a lot in the automotive industry. But an H-bride is no rocket science, and building it with discrete power transistors is very doable too. And this avoids availability problems of highly integrated chips. Those highly integrated chips are also not much cheaper in low quantities.
It's probably doable to connect all motor drivers to a single microcontroller, but using a single uC for each motor driver, and then yet another system that does the higher level math and controls all the motor drivers is a more logical approach. Microcontrollers are so cheap that it does not make much sense to put everything in one uC. I do find it sort of impressive that the "Hiwonder" PCB you found can control 4 motors. It's max motor voltage is apparently 15V, so that is enough, but 2A can be a bit low for your motors, and that would be a limitation. You want high peak currents for quick acelleration. What are the specifications of the motors in the Scorbot?
You can also have a look at GRBL (Now: GRBL HAL) It spits out step and direction pulses made for stepper motor based systems, but it is also relatively common for motor drivers for DC motors to accept this as their input.
LinuxCNC is also an option. It's a very flexible system. It can do custom kinematics and also control a lot of motors, but it also has a quite steep learning curve.