The main distinction here is between global and gauge anomalies.
As you described, a symmetry is anomalous if it is realized by our classical description of the system but is broken quantum mechanically (i.e. the Lagrangian is invariant under the symmetry but the path integral is not).
Generally, it depends on the system you're considering and what type of symmetry is anomalous, but I’ll focus on anomalies in particle physics. Roughly speaking, we should distinguish between anomalies in global symmetries and in gauge symmetries. As you suggested, we can measure global anomalies in particle physics by observing some decay process not allowed by that symmetry (e.g. proton decay and baryon number conservation).
These are distinct from gauge anomalies. Gauge symmetries are not real symmetries of the physical system but are redundancies in our description of the theory (which we put in to make, for instance, Lorentz invariance manifest). Because gauge symmetries are not real, they shouldn’t be broken by quantum effects. This is why gauge and gravitational anomalies must all cancel in the theory, otherwise the quantum theory would be inconsistent. Gravitational anomalies are simply another kind of gauge anomaly, as local diffeomorphism invariance is a gauge symmetry of gravity.
So global anomalies are fine, the quantum theory just didn’t have some symmetry you thought it might, and we can measure these in experiments. Gauge anomalies (and gravitational anomalies) are a sign that the theory doesn’t make sense and should be zero in a physical system.
A few other things to note:
- There are a number of other interesting anomalies not mentioned above, e.g. 't Hooft anomalies.
- Conformal anomalies indicate the breaking of the scale invariance of a theory. QCD (with massless quarks and gluons) is classically scale invariant, but we know that this must be broken as QCD confines (and the quarks and gluons around us sit inside protons and neutrons).
- Technically speaking, a gauge anomaly sometimes means that you need to add more degrees of freedom to the theory.