Nonuniform circuit families - don't have to specify for arbitrarily large, but finite, input lengths?
Although families of circuits aren't usually defined this way, you actually could require this for non-uniform families of circuits; i.e., require circuit families to be consistent. There still would be circuit families for non-uniform languages, and P/poly would still contain all the same languages with this definition (P/log would change, though). Then you could indeed leave out any finite number of circuits, and the function would still be well-defined.
You wouldn't cause any paradoxes.

One question you could ask is: are there any advantages to this alternate definition of P/log? My guess is that there are.