IMHO, both local calibrations (i.e. fitting to specific instruments) and global calibration (fitting to all swaptions simultaneously) are valid approaches and I've seen them used on a trading desk.
Local calibration is (generally) easier to implement and allows you to reprice the instruments you are likely to use as hedges, with more accuracy than a global fit. Using different calibration sets for different trades means you are effectively using different models for different trades. Or as DavidJN puts it, you are using different models to describe different parts of the curve. There are endless debates on whether this is a good thing or not.
A global fit on the other hand, with a suitably high dimensional (e.g. 3 or more factors) should be able to capture the main dynamics of the curve, and be fitted to the entire swaption matrix. The degree of fitting success depends on the specification of the volatility function you use in your term structure model (e.g. separable, time-homogenous, etc), correlation structure (related to factor loadings in a LMM model) and the numerical scheme you use - e.g. global optimisation, cascade calibration, etc.
I would guess that for insurance purposes, having an exact fit to all parts of the swaption surface is not as important as capturing the overall curve dynamics, especially if the resultant model is to be used for generating rates scenarios. I would not dismiss global calibration for this purpose.