I'm not going to comment on the airworthiness of Gyros per ce, since there are others far better equipped to do so than I am.
But it is clear that in the UK, the gyroplane safety record is quite horrendous - sat around 1 fatal per 8,000 hours. I'd like to venture an opinion of why that is.
If you look at the UK gyroplane community it is small - around 250 aircraft, of which maybe half are in flying condition. This is spread around maybe half a dozen clubs, so there are perhaps 20 flyers in a typical club - and this is right across the country, so they aren't in very regular communication with each other.
Also you've got initial approvals being done by the CAA, in-service airworthiness by the PFA, pilot training oversight by another bit of the CAA; in other words various components of the "system" again not in routine communication with each other.
So the result, in my opinion, is that safety lessons - be they operational, design or maintenance are not being circulated and learned from. The gyroplane community in Britain almost certainly need to be far closer together than they are at the moment, with mechanisms for establishing and ensuring best practice set up and stuck to.
Once you've done that, any deeper issues with gyro safety can probably then start to be dealt with, but at the moment the real problems (in my opinion) aren't really with the basic design of the gyroplane.
G