There are a few reasons I have safety concerns about Tiger Moths being used to carry members of the public:
1) fuel tank above an open cockpit, also any overflow from the fuel tank is likely to spill into the cockpit.
2) lack of brakes in the original design.
3) lack of an electrical system in the original design.
4) most Tigers seem to have had upgrades for brakes, engine and avionics, but this seems to have been done on an adhoc rather than systematic manner that I suspect was never envisaged when the original Type Certificate was issued.
5) marginal performance, that appears to rely on pilot skill rather than systematic design.
6) my opinion is that the near-perfect airline safety record in Australia gives the public a perception that flying is systematically safe and that they would be surprised that aircraft owners are relying of 70 year old certifications to fly their vintage planes today.
7) accident history.
My opinion is that maybe it's time to move vintage aircraft like Tiger Moths off the Normal/Airwork/Charter register and into Limited or even Experimental.