have high pressure on their top and right, low pressure on their bottom and left. So in this case, air must go left
Your observation is correct. The moving fan creates a high pressure zone behind it and a low pressure zone in front of it. However your conclusion is wrong. It is not a "must" that air go to the left.
Consider a balloon (just a regular one, made of rubber and inflated by human breath). The air pressure inside the balloon is higher than the air pressure outside the balloon. Therefore by your reasoning you may think that air must move from the inside of the balloon to the outside. But this does not happen. The balloon stays inflated. Why is that? It's because there is a wall separating the inside of the balloon from the outside - the rubber skin.
Exactly the same thing happens with the fan blade. The movement of the fan creates a zone of high pressure behind it and a zone of low pressure in front of it. But the high pressure air does not flow to the low pressure zone for exactly the same reason - there is a literal wall separating them: the fan blades.
But you say, "look: there are gaps between the blades"! Yes, and some air can move through the gaps. However air has mass and therefore inertia. Air cannot instantaneously teleport from one location to another. By the time the air tries to move into that open gap the fan blade would have moved to cover it (but obviously leaving a gap behind it - then the air may try to move into that gap but by the time it can get there the fan blade from behind will move to cover that gap (leaving a gap in front of it but again it takes time for air to move and by the time it can get to that gap the fan blade would advance to cover the gap... repeat to infinity)).
Now, if this is all that's happening sooner or later the two sides will sort of equalize anyway and there would be no pressure difference. All the fan would do is create turbulence in the local space with air bleeding through to the low pressure side via gaps we can't quite cover (eg. the gap between the fan tips and the engine's duct/body). You can see this phenomena happening in blenders. However, thinking this way ignores an important fact: the fan is not the only object in this universe - the fan is operating in a whole universe that surrounds it.
One thing you are missing from your thinking is that the entire engine is surrounded by the planet's atmosphere. Yes, there is a low pressure zone in front of the fan blades but you are forgetting that there is an atmospheric pressure zone in front of that zone. Since air moves from high pressure to low pressure and since the atmospheric pressure is higher than the zone in front of the blades air will move from the outside world into the engine. This is the sucking action.
It's the same for air behind the blades: yes there is a high pressure zone behind the blade but there is an atmospheric pressure zone behind that zone. So air will move form that high pressure zone to the low pressure of the atmosphere behind it. This is the blowing action.
But how do the blades themselves create the high pressure and low pressure zones? The same way aircraft wings do it (or your cupped hand do it when you put your hand out the window of a moving car). The aerodynamic effect of the blade's airfoil coupled with the blade's angle of attack creates the pressure differential. We normally call this effect "lift" but in fans we call it "thrust".
And yes, as I mentioned earlier this lift/thrust is a transient effect (you can think of it as the blade beating down on the air). If left alone the air would move back to their original positions and thus no net air movement would be created. But the blade moves continuously, constantly creating the pressure differential and not giving the air a chance to recover back to equilibrium. It's kind of like the difference between swinging your hand and putting your hand out of a moving car. When you swing your hand you do get lift but since you stop swinging at the end you don't even have time to feel that lift. Putting your hand out of a moving car however you can feel the lift because your hand is constantly moving into fresh air and thus never allowing the high pressure zone below your hand to equalize with the low pressure zone above your hand. It is the constant movement that forces the two zones to remain separate.
If the engine were to stop spinning you will find that the high pressure zone and low pressure zone will disappear. This may sound kind of obvious but they disappear not just because the engine is off. They disappear because once the fan blades stop moving the air from the high pressure zone can equalize with the low pressure zone because there is no more moving fan blades stopping it from doing so.