"PAT testing has without a doubt saved lives, prevented fires" (My emphasis)
For the most part, no. PAT testing is largely worthless because the testing involved only detects for conditions likely to cause electrical shocks to humans, conditions which are relatively rare given most mains devices thesedays are a switched mode power supply which then provides an isolated lower voltage to the stuff which actually powers everything the device does.
A device can easily have a multitude of internal faults which could lead to high current draw and fire, a PAT test wouldn't spot them. A fuse should stop some of those though, a superiority of the UK plug design over countries which have fuseless plugs. But even then, a device can consume a current within its fuse rating, and concentrate enough heat within it at some particular location to get hot enough to set itself alight.
Add Lithium batteries in to the picture, now contained in so many devices, and you're often looking at non-mains devices being more dangerous than mains ones.
A PAT test also makes very little sense when a device has internal states which can change and control higher powered parts of it, via relays or such, which mean the exact configuration of the device during a test might be quite different to what a device might have when in use, because perhaps the device switches some functionality on after 5 minutes, or when an internal temperature passes a threshold... The test gives no indication of what the device may do in that state.
And such tests on a device, which merely prove shock is unlikely, not that is safe from fire and such, only show it passed when the test was done. Any accident with someting faulty is likely to happen immediately after the fault develops, the chance of a fault developing, then being spotted in a test before the fault has a harmful consequence is negligible.
Checking something looks in good condtion, and is from a decent manufacturer makes sense, but the formal tests are frankly focused around the designs of devices from a historic era, and overly focused on shock a relatively minor risk (worse case person holding it dies) as versus fire a much worse risk (the whole building burns down and multiple peope die).