No Stupid Questions

Here is an easier example.

Think of a truck that sits on a street that is going slightly uphill. There is no way that you and your buddy could lift that truck straight up in the air. But you can relativily easy push it uphill. In the end the truck is going up.

Now, what's the physics behind that and how does it relate to planes? Well, you don't have to lift the entire car when pushing. Most of the force from gravity is resisted by the ground. You only have to push against the much smaller horizontal component that tries to push the car downhill.

With planes you basically just replace "ground" by "lift". So instead of tires pressing against the ground, you have wings pressing against the air. And a jet engine instead of two guys pushing.

Basically a plane's engines are pushing the plane up a hill made of air.

This part is half-right: the work of the engine is going to accelerating the plane forward, or (when thrust and drag are in equilibrium) maintaining the current velocity. But in level flight, the engine is not "keeping the plane in the air" - it is impossible for it to contribute to lift directly because it's force vector is 90 degrees from the lift vector.

Therefore, some of the engine’s energy is going into keeping the plane in the air, and some is going into accelerating it forwards, or keeping it at the same speed (fighting air resistance).

This is where you make an unsupported leap:

Therefore, if the plane points straight up, the engine should be able to support it hovering in the air. If it didn’t have enough power to fight gravity when pointing straight up, it wouldn’t have enough power to fight gravity when moving horizontally, either.

A car can accelerate horizontally because its engine can rotate its tires to apply horizontal force due to friction and mechanical advantage; does that mean it can drive straight up a wall? Of course not (outside of some specialized bouldering vehicles). The engine lacks the power to lift the car straight up, and the tires lack the grip to hold on to a vertical surface. The drivetrain is designed for efficient road cruising, not high power and grip

It's the same for aircraft, generally: a given engine usually has enough power to accelerate the aircraft horizontally, and applies this through some kind of prop or jet rotor. But this combination is tuned for efficient cruising, not vertical climbing. The engine won't provide enough power, and the prop can't move enough air, to sustain vertical flight indefinitely.

"But Sleet01," you cry, "helicopters exist!" Just so! They trade cruise efficiency for vertical thrust by greatly increasing the size of the prop, increasing the mechanical advantage so that less engine power is needed to hover or climb vertically. That's like putting 4" wheels covered in suction cups on your car - now it can go straight up, but you can't go very far or very fast!

"But Sleet01," you exclaim, "F-15s exist and can fly vertically almost to space!" Indeed they do, but in order to fly an F-15 vertically you need to burn immense amounts of fuel, almost 400 gallons per minute. That's like putting two turbo V8s in your Jeep - now you have the power to go vertical, but only for a couple minutes!