One pilot told me years ago: “ the 747 has all the gliding properties of a brick.”
From what I have ever seen,major jet catastrophes were pilot error,weather and tail/rudder issue. Catastrophic engine is very rare. Who am I? I know internet!
Once a 747 loses 1 engine its ability to maintain altitude is compromised and it will have to descend, however, it can continue at a lower altitude. A BA 747 took off from Los Angeles some years ago, lost an engine very soon after take off and continued all the way across the Atlantic at a lower altitude than planned. However, although only 3 engines were in use they used more fuel than the 4 would have at a higher altitude and they diverted to Manchester instead of continuing to London.
Lose 2 engines and things get much more interesting. If the two non functioning engines are both on the same side then the aircraft will tend to yaw significantly. The aircraft wont be able to maintain its cruise altitude but by descending to a lower altitude control can be maintained and if the altitude is low enough the aircraft may even be able to climb very slowly. An immediate divert will be called for.
Lose 3 engines and you will have problems maintaining altitude. The one engine remaining will help to slow the altitude loss but you WILL lose altitude. Another BA incident involved a 747 that flew into volcanic dust over Indonesia. They lost all 4 engines all of which were contaminated by the dust. They got one going at about 12/13000 feet which helped them to reduce the rate of descent but only when a second was restarted could they begin to climb again but it was painfully slow.
So, can a 747 fly with one functioning engine? Yes. How? By maintaining a slow rate of descent (to the crash site).