I agree with you that if a new theory was to replace the old one for making specific set of predictions, it should give predictions of similar or better accuracy. But I do not think <i>that replacement</i> is necessary for the new theory to compete or be accepted; it is the new benefit it brings, whatever its nature may be, that is crucial. The two can temporarily both be accepted to coexist, if both have their strengths. For example, quantum theory does not make the same predictions as classical theory when it comes to classical experiments (mechanics, basic EM phenomena) and is largely useless in that domain. It only gives probabilities of results of specified experiments of certain kinds; it does not reproduce the old predictions (like definite trajectories, Moon phases or solar eclipses), but provides new results (like resonance frequencies of atoms and molecules and their bond energies). Similar thing can happen with a new theory of gravity; it may not give the same prediction for Mercury perihelion precession, but it may be able to explain other things, like why the inverse square law, why no repulsive gravity or why the mutual gravity force between electrons is so much lower than the mutual EM force. Explanation for oddities in Mercury motion could then wait for further data and repetition of calculations. It is natural to expect of any new theory to bring new results, but demanding that it reproduces all the old ones along is too much. That happens rarely and such expectation only prevents any new ideas from being considered.