I want to build a camera rig for still shots with a monitor and 1 or 2 flashes. Such rig would force me to have 3 different types of batteries and I don't like this complexity. I wonder if there are good solutions for having one big power-source (let's say a power bank) and make it supply power for the camera, flash(es), and monitor?
Your requirements are very specific so something handmade will be more appropriate for you. You can use mobile car starter like this which provide 12V and good amount of ampers. Then using DC-DC convertors you can get the voltage you need for different devices (camera, flash, monitor). And then dummy batteries for the units.
All this will be avg. 1.2-1,3kg so you can carry it around.
The single power source is a solved problem. Professional video cameras use V-mount or Gold-mount batteries. They are available with taps that can power most contemporary DSLR and Mirrorless cameras as well as common video accessories like monitors and small continuous lights. Camera cages, rails, cheese plates etc. provide a vast combination of rigging options that accept battery mounts.
The only non-obvious challenge is powering camera flashes, since external power sources are usually of a proprietary design and the input voltages may be different from standard video battery taps. On the other hand, the requirements of a flash are fairly basic electrically and AAA battery powered flashes are common.