Not all gauge transformations can be generated by infinitesimal gauge transformations in general

http://eduardo.physics.illinois.edu/phys582/582-chapter9.pdf
9
Quantization of Gauge Fields
We will now turn to the problem of the quantization of gauge theories. We will begin with the simplest gauge theory, the free electromagnetic field. This is an abelian gauge theory. After that we will discuss at length the quantization of non-abelian gauge fields. Unlike abelian theories, such as the free electromagnetic field, even in the absence of matter fields non-abelian gauge theories are not free fields and have highly non-trivial dynamics.
9.1 Canonical Quantization of the Free Electromagnetic Field
Maxwell’s theory was the first field theory to be quantized. However, the quantization procedure involves a number of subtleties not shared by the other problems that we have considered so far. The issue is the fact that this theory has a local gauge invariance. Unlike systems which only have global symmetries, not all the classical configurations of vector potentials represent physically distinct states. It could be argued that one should abandon the picture based on the vector potential and go back to a picture based on electric and magnetic fields instead. However, there is no local Lagrangian that can describe the time evolution of the system now. Furthermore is not clear which fields, E or B (or some other field) plays the role of coordinates and which can play the role of momenta. For that reason, one sticks to the Lagrangian formulation with the vector potential Aµ as its independent coordinate-like variable.