But only if the character's Int is 14 or greater (or 16, if one goes with the more conservative modifier), which is not always feasible or even desirable. And it represents an opportunity cost. For a Human Fighter (the smallest pool with the greatest number of trained skills), an Int of 14 would net them an additional class skill. But what would he give up in exchange for training this skill?
Even so, there's more to skills than just using the ones you're trained in. Adopting this house rule imples a certain skill consciousness on the part of the DM, I suppose, in that you will want to test your players even in skills they don't have training in. But I can't see that it would undermine skill training entirely unless the DM were asleep at the wheel.