There are two types of starter dogs (which engage with the flywheel starter ring gear.
Modern ones are what we call ‘pre-engaged’ - where the starter motor dog is engaged mechanically/electrically by a solenoid before the stater motor windings are contacted to transmit turning motion. This type remains in mesh untill the starter is de-energised by the operator.
The typical Moggie Minor starter is of the older type which engages the starter dog by throwing it forward to engage with the starter ring. It will only remain in mesh while it is driving the starter ring. As soon as the flywheel removes the ‘grip’ the bendix (starter dog arrangement) withdraws from the starter ring (while the starter motor speeds on doing nothing) - that is what you are hearing. Generally the starter disengages when the engine fires and starts.
In your situation, there appears to be only just enough flywheel acceleration, from one weak cylinder firing, to allow the starter dog to disengage, but without the other cylinders firing to start the engine.
I would, at this stage, remove the plugs, clean as necessary and then dry them with a gas ring on our cooker

, replacing while still hot (minding burning fingers or dropping them) and try the starter again immediately. Usually works.
That plug you removed is awful. The engine was either running rich for some time or the engine is burning oil (most likely the latter, or both).