Why local transformations must be gauge transformations:
Traditionally, quantization is recipe in which the phase space of a classical system is replaced by a Hilbert space of a quantum system; and functions on the phase space representing the observables are replaced by operators on the Hilbert space. Also, the action of the classical observables on the phase space is replaced by a quantum action of their quantum counterparts weighted by a parameter $\hbar$ such that in the limit $\hbar \rightarrow 0$, the action coincides with the classical action (correspondence principle).
Even though in many applications it is not explicitly pronounced, a quantization procedure should start from a phase space. The basic meaning of a phase space is the space of all possible initial conditions (space of initial data). On the basic level, we deal with systems whose equations of motion satisfy the property of existence and uniqueness of solutions; thus each initial condition corresponds to a unique solution. Therefore we can think of the phase space as the space of all classical solutions. The latter definition of phase-space has advantages as it doesn't need to separate the time from the other coordinates and allows a covariant definition of the phase space. In the physical literature, it is known by the Crnković-Witten formalism.
When local symmetries exist, the property of uniqueness of solutions is lost and there are combinations of coordinates or fields in the Lagrangian which are not controlled by the equations of motion and can assume arbitrary values. The theory cannot say anything about them. On the other hand, the combinations which are controlled are exactly the gauge invariant combinations. This is one of the consequences of Noether's second theorem.
Remembering the basic definition of the phase space as the space of initial conditions; the best we can do is to work with the subspace of initial data which the theory can control; i.e. the space of gauge invariant observables. These observables generate the reduced phase space, i.e. a phase space in which the local symmetry is gauged away.
This space in general is not a manifold. It contains points of singularity, which make it hard to quantize even is simple quantum mechanical systems; please see Emmrich and Römer. This is why methods like BRST are used to impose the gauge symmetry after quantization.
On the Lattice:
On the lattice, only correlators of gauge invariant observables are computed. In this case the gauge redundancy is manifested by a multiplicative constant of the volume of the discretized gauge group in both the denominator and the numerator. We would be committing an error if we had computed correlators of gauge non-invariant quantities which are not controlled by the theory. Their correlators will not depend on any parameter of the theory (such as coupling constants) that we want to study and they would produce just a random result, very sensitive to the method that we chose to interpret them as quantum observables.
Also, it is much more convenient to work on unreduced phase space on the lattice. It would be extremely hard if we had worked on the reduced phase space which as explained above a very complicated space.
In contrast to the case of gauge symmetry, the theory does not constrain how we treat global symmetries. As long as they are not anomalous, we have, in principle, the liberty to gauge away global symmetries or leave them as symmetries of the system (classical and quantum). In the first case we interpret configuration related by a symmetry operations as the same physical state, while in the second, we interpret them as distinct physical states related by symmetry. A special case of these symmetries is the large gauge symmetries which in many examples act as symmetries and not redundancies as they connect between physically distinct states. This subject was discussed many times here in Physics stack exchange; please see the following answer and the references therein.
Asymptotic symmetries are "gauge" symmetries on noncompact spaces or spaces with a special closed surface leaving the boundary conditions invariant modulo those connected to the identity component. These symmetries generate, in certain cases infinite dimensional Lie groups, are also not included in Noether's second theorem which assumes compact support of the variation. These symmetries give rise to physical charges such as electric and magnetic charges. Hence, they should also be considered as global symmetries from the point of view of the quantization process.
Asymptotically trivial local symmetries must be considered as gauge transformations