Why do airplanes not have manual transmission?
Aircraft have fixed gears (when they have gears at all)! However in a sense a variable-pitch propeller can be considered analogous to transmission in a car and it was manual in some aircraft.

The power transmitted by propeller is, within some reasonable range, proportional to its rotation rate. And so is, again within some reasonable range, the power produced by the engine. So for slow planes a fixed pitch propeller with connected by fixed gear does a decent job. As you open the throttle, the engine speed increases, so does the propeller and that's exactly what it needs to do to transmit the power. Simpler aircraft have no gears at all and propeller is mounted directly on the engine shaft, higher power engines often have 2:1 reduction gear as the engine rpm would be too high for a propeller.

Now at higher speed, the apparent propeller pitch is reduced as the air is already moving, so it would spin faster and faster and eventually too fast for the engine. To counter this, faster propeller-driven aircraft have variable pitch propeller. As speed increases, it is set to higher angle of attack (coarser) so it keeps delivering power without increasing rpm.

Early variable pitch propellers had manually controlled pitch, but quite quickly constant speed propellers were developed (many WWII aircraft had them already) where a weight-on-spring controls the propeller pitch so that constant rpm is maintained. The manually controlled units required a lot of attention and the constant-speed system is rather simple, so manual pitch control was quickly abandoned.

And that is kind of equivalent to perfect variable gear transmission. The pilot still has a throttle lever and a propeller pitch lever and selects power and rpm independently. Higher rpm allow higher power, but don't require it, so landing is usually done with low power, but max rpm so the engine reacts quickly if power is added (does not need to spin up). During cruise some more moderate value is selected (according to manufacturer recommendation) to reduce wear of the engine. 

Turboprop engines always have constant-speed propellers, because in a turbine engine the difference between turbine speed at idle and at full power is not actually that big. Again the rpm limits maximum power, so landing is done with low power, but high rpm and the engine reacts quickly when power is added.

Last the turbojet/turbofan engines just turn the turbines and compressors as they need. It's not much different from the fixed propeller really, though the jet engines now usually have variable guide vanes in the stator (that are just linked to rpm).