What is the actual question you’re asking here?

If you’re asking about just the mechanics of a Microwarpdrive, they grant you a percentage thrust bonus and blooms your signature radius while it is cycling.

If you’re asking about “why” a ship is able to enter warp with a 1 cycle of a prop mod (this applies to both Afterburners and Microwarpdrives), it’s because of how warp thresholds interact with your ship’s current speed.

All ships in EVE enter warp when they reach the 75% maximum velocity threshold. This means that if your ship has a maximum velocity of 100m/s, and you issue a warp command, your ship will begin accelerating from 0 m/s until it hits 75m/s, at which point you will enter warp.

Turning on a prop mod temporarily increases your maximum velocity. For the sake of numbers, let’s say your ship’s default max speed is 100m/s. And a prop mod increases that by 100% (basically doubling your ship’s max speed). By turning on the prop mod, your maximum speed is now 200m/s.

Now, in order to enter warp, you ship needs to hit 75% of the NEW maximum speed, namely, 150 m/s. So your ship will begin to align and start picking up speed, it’ll pass the original 75m/s warp threshold before you turned on the prop mod and will continue to 150 m/s because that is the new 75% max speed threshold.

But what happens if you turn the module off?

Your ship maybe still be picking up speed, in fact, it may be around 120m/s when you turn off the module. As soon as the module is off, you no longer gain that percentage max speed bonus (the 100% in our hypothetical example), and so, your ship’s maximum speed drops back down to it’s original amount, 100m/s.

Your ship is now fully aligned, going 120m/s, when its maximum speed is 100m/s. 120m/s is clearly greater than the 75% max speed threshold you need in order to enter warp. Thus, the game sees that you have satisfied this requirement and allows your ship to enter into warp.

In game, there are several additional variables that go into play, such as your ship’s default mass, the additional mass added by the prop mod, the percentage thrust bonuses, etc.

I don’t see why a ship should be “required” to fit any specific type of propulsion module. Capital ships can enter into warp with a single cycle of a 500mn MWD, which is normally considered to be a Battleship prop mod.