It certainly simplifies things to use 0 instead of 1. It's just that it seems odd to me to use 0 when the coordinates are defined as being factors of the number they represent.
I feel that by using 0's, I can represent lower-dimensional objects in higher dimensions (0 just means that they don't exist in that dimension), but by using 1's, I can really create higher-dimensional objects, which seems to me a more satisfactory way of expressing a bunch of numbers in the same dimensions. Unfortunately, I have no idea why this approach would be useful at all! But it's always fun to play with numbers and try different things.
The problem with 1s is that as you change the dimension, you change the numbers. The problems with finitely many dimensions is that you can't represent all numbers. Using infinitely-many dimensions also solves another problem: the arbitrariness of which numbers go in which dimension. With infinitely many you can have a dimension set aside for 2, 3, 4, 5, 7, 8, 9, 11, ... so that all numbers divisible by 3 have the same coordinate in that dimension.