Indeed, that is a big impediment in many disciplines. Non-perturbative QCD (i.e. what describes the mass around you) can not be calculated analytically with a c

Indeed, that is a big impediment in many disciplines. Non-perturbative QCD (i.e. what describes the mass around you) can not be calculated analytically with a controlled level of precision. The best we can do is lattice QCD, i.e. the brute force approach. That's a big difference to QED, where we know how to find better and better approximations in a systematic way. In other words, QCD, like many complex systems, shows emergent behavior not easy to predict from the theory input (couplings, masses). A better mathematical framework to attack these kind of problems would be a substantial jump forward.