You are correct : internal forces do not accelerate a system. This is because all forces occur as action-reaction pairs, and every internal force is paired with another internal force. Each force in this pair acts on a different part of the system, so there is no net force on the system.
It is only external forces, which some other object exerts on the system, which can accelerate a system. The pair to an external force is a force exerted by some part of the system on an external object outside of the system. So the effect of the external force on the system is not cancelled out by its pair.
BEWARE! Some websites have a very different 'definition' of what internal vs external forces are - eg The Physics Classroom. A similar situation to your "real forces" question.
Using your example of a painter :
If "the system" consists of the man and the platform but not the rope, then it is the upward pull of the rope on the man and on the platform which (after subtracting the weight $mg$ of the system) accelerate the system upwards :
$ma = 3T-mg$.
These 3 forces $T$ - coloured blue in the illustration - are external forces.
Note that the downward pull of the man on the rope (not illustrated) is not the force which accelerates the system (man and platform), because (i) it acts on the rope, not the system, and (ii) it acts downwards, whereas the acceleration is upwards.
If "the system" includes the rope, then the pull $T$ of the rope on the man is now an internal force, and does not accelerate the system. The external forces which accelerate the system upwards are now the force $2T$ which the uppermost pulley exerts on the rope bent around it, and the force $T$ from the ring-bolt P acting on the end of the rope which is tied to it. These forces are marked in red in the illustration.
This seems strange, because the objects exerting these forces (the upper pulley, the ring-bolt) do not move upwards themselves. The point at which the force is applied does not move - so how can these forces do work on the system?!