The problem would still exist, for example when you want to multiply colors which is common in lighting calculations - with integers you would need to divide by either 255 or 256 after such an operation. Which floats you only need to make that decision when converting to/from float representation, which is part of what makes them convenient.

It is because light(colors) is a fundamentally exponential process and floats are also a fundamentally exponential process and as such the two are a good match for one another.

Floats are better for calculating lighting. I would think some GPUs are probably also more optimized for float processing than ints.