That argument would apply equally well to discarding FPTP and retaining a monarch (or a dictator) at the top. A strong leader is definitely stable, so if that's the key metric an FPTP system is still not a winner.
PR certainly increases the power of minorities (but only in proportion to their support among voters) vs a FPTP system in which those voters are not represented at all; this can be both positive or negative (if NZ's system was implemented here in Australia, both the Greens and One Nation would have far more representation here, and the two major parties would be far weaker).
I'm not convinced that this is a bad thing. Where there is no representation people may justifiably resort to other means and that's hardly stable either. NZ's experience with MMP has been largely positive.
> It is truly bizarre that this is always one of the things that people recommend when it is well understood it would have the opposite effect from intended.
It may be that it's often recommended because this is not necessarily the case at all; however, I don't know the specifics of the UK setup and so whether it would be helpful _now_.
I think it's hard to argue that a less representative system is not also less democratic, so hopefully whatever is holding you back over there can be sorted out somewhere down the line.