I’ll give it a shot - I’m sure if I get any details wrong someone will be along to correct me shortly.
While the rotation of fan blades does have some effect, it is not the primary one
In a configuration typical of most airliners and multi-engine airplanes, when you have an engine (or two) on each wing, assuming the engines are all producing equal thrust the airplane is being pushed along in a stable manner. The axis of the thrust is lined up with an axis of the airplane and the axis of travel. Very nice and neat. Now, if the right engine quits (for example) that wing isn’t being pushed along anymore, but the left one is still going forward at the same speed. Only problem is, it’s still attached to the right wing, which is going slower (not being pushed and all). The right wing lags, the left goes forward, but, being achored to the slower wing, the airplane veers right. This is called “asymmetrical thrust”, To correct usually requires application of rudder, lots of it, to counteract the right-turning tendency. Reverse left and right in the above for the sceanario with a left engine failure.
(A very somewhat sort-of analogous thing would be a tire going flat in a car – it causes more drag on that side, and the car wants to veer towards the flat.)
Now, one engine or the other will be the “critical engine”. What that means, basically, is that if the critical engine quits it will be more difficult to control the airplane than if the non-critical engine quit. And this is where the direction of fanblade rotation (or propellor rotation), and the airflow over the airplane, and a bunch of other factors come into play. Design can compensate somewhat for this (for instance, the closer the engines are to the centerline of the plane the less assymetrical the thurst will be with an engine not functioning), but not eliminate the problem completely.
With something like a Cessna Skymaster, which is called “centerline thrust” because both engines align with the longitudinal axis of the airplane, the problem is slightly different. No matter how many engines are working or not working, the thrust generated is always symmetrical. But, the front of the airplane is a different environment than the rear. Airflow over the fuselage has an effect, for one thing. Also, the stronger the airflow over the tail, the more effective the rudder and elevator are. In a Skymaster, the rear engine thrust is very close to the tail, but if that engine quits you have only the front engine… and there will be a noticable loss of effectiveness in the rudder and elevators. Not fatal or unmanagable - but noticable. If I recall, there are some business jets that use centerline thrust and don’t have a tail assembly behind the rear engine (probably because the exhaust would melt it). I presume they, too, would have some issues with single-engine operations, but that’s not an area I have any expertise in at all, so I won’t comment on those.
Now, a word or two about slow speeds and airplanes. Airplanes are like sharks - they have to keep moving or die. It’s the air going over the wing that generates the lift that holds you up, and you need that air going at a certain minimum speed to generate the minimum lift required to hold you up. (When no engines are running, you need maintain that forward speed, too, and you do so by a controlled descent. Gravity provides the motive power in that case, the trick is to release the potential energy slowly) With engines operating normally, you can generate more than the minimum, which is how you go up.
Now, the control surfaces - ailerons, rudder, elevator, etc - are mostly airfoils themselves, or modify the shape of the airfoil they’re attached to. So they, too, require a certain minimum airspeed in order to work.
So here’s the hitch. At cruising speed you have LOTS of air going over all those surfaces and everything works very nicely and easily. But when you slow down, you have less air and less lift, so nothing is quite as effective as before. If you want to turn left, for instance, it will require more input into the controls than if you were in cruising flight. Again, in normal flight this is not going to turn into a problem - it’s like the difference in handling between your car in first gear and in fourth.
However, if you get so slow that you’re right on the edge of having enough lift to keep you up, all the controls are going to be on the edge of effective/not effective. It’s somewhat like driving a car on an icy road (airplanes also have the added feature that, in addition to the sloppy feeling in the steering, most of them also tend to vibrate and shake at this point, too. This is good - it helps get the pilot’s attention.)
OK, back to normal take-off. At take-off, the name of the game is to get away from the ground. Why? because if an engine quits, the only way to save your butt is to trade altitude for the airspeed you need to keep the controls effective so you land instead of splat. And to do that, you need altitude. The more the better. What’s altitude? It’s how far away from the ground you are. So you divert most of the engine power to the UP force and less to the FORWARD force. And works just fine – unless your engines quit. Then you’ll have much much less UP and FORWARD. If the UP and FORWARD runs out before you can reconfigure the airplane for a glide, it can hurt really really bad (but, on the bright side, only for a moment…) In take-off configuration, you don’t have much extra airspeed. Which is why you MUST react quickly.
(Contrast this to landing, where you’re using minimal engine power. In fact, the ideal on landing is to have the engines at idle, so you’re essentially gliding in. If the engine quits you’re already in glide mode, which makes the situation less critical at least in regards to airspeed.)
Tired yet? Don’t worry, I’m almost done.
OK, back to our assymetrical thrust sceanario (that means, at least one engine has quit). Alright, we’re taking off - going up - but we’re still low, and we’re relatively slow. Suddenly, half the UP and FORWARD disappears. The plane yanks itself to one side. (Think “blow out of tire on freeway with engine sputtering and coughing”) You have to, above all else, make sure you maintain enough FORWARD to keep your airspeed high enough to generate enough lift to keep you in the air and under control. The only way to do that is to divert some of the UP force to forward - which means you can’t climb very much, if at all. Whatever altitude you’re at — that’s where you’re most likely going to stay, or lower. Meanwhile, because the remaining engine wants to pull to one side, what thrust you do have left is not being used efficiently - that thrust pushing you to one side of directly forward is being wasted from your viewpoint. AND you have to put in a lot of rudder, which increases the drag slightly (normally not a problem, but you’re on the edge here), which further decreases efficiency. If you were way high up, you could lose a couple hundred feet - or even a couple of thousand feet - of altitude in order to make sure everything was set up proper and in control before refining the airspeed for maximum efficiency, and the trade would be a bargain. But close to the ground you have no altitude to trade.
Start to get the idea? A pro can make it look easy, but it does take some skill. And lower you are, the more difficult it is.
In fact, the minimum controllable airspeed (that’s just what it is - the minimum speed at which the airplane can still continue in flight) for a multi-engine airplane with just one engine is significantly higher than the airspeed for two engines. Because assymetrical thrust is so much less efficient.
Thus, standard procedure is to maintain the airspeed at or above “one-engine” speed at all times.
And any damage to the airplane will increase that minimum controllable airspeed. In order to land a heavily damaged airplane the pilot must fly it to the field at a higher than normal speed. If the damage is too great, the drag will be too much, the lift too little… and the consequences severe.
Thus, we return to the reply to the OP that’s been stated several times - survival depends on what damage, other than a destroyed engine, also occurs.