There are a few things to clarify:
It is very important in your question to differentiate between the strong force and the residual strong force (nuclear). The reason is, because you are asking about the nuclear force, mediated by heavy mesons. Though the strong force itself is mediated by massless gluons inside the nucleons.
the rest mass of the mediator (heavy mesons) makes the residual strong force (nuclear) short ranged, just like the weak force. This rest mass does not have anything to do with the relative strength of the residual strong force. Now the short range of this force is a little bit complicated, because the residual strong force is repulsive at very short distances. This keeps the neutrons and protons apart. At 0.8fm, the nuclear force is very strong attractive, and at 1-3fm, it is still strong attractive.
it is the coupling constant that determines the relative strength of the strong force. It just happens so, that the strong force is way stronger then any other force at short distances (under the size of the nucleus). Now this is not precisely correct, because the strong force is very strong under 0.8fm (the inside of the nucleon, nothing can come close to it), and is strong at 1-3fm (nucleus), but at very big nucleuses, with over 200 neutrons and protons, the EM force starts to overcome the strong force.
Now you are asking about virtual particles. when we create mathematical models, we try to explain in math the data coming from experiments. These data fit perfectly our currently accepted theories, the SM, together with QM and QFT. When we try to model these interactions between the field (in this case the residual strong field), we use virtual mesons and these are not real particles.
Yes these virtual mesons do have rest mass (off shell), and are not real particles, they are just a mathematical model.
You are asking why the strong force is stronger then the EM force. In reality we do not know why. We just accept that the universe is built so, and built mathematical models for the strong force and the EM force, and the coupling constant happens to be 137.