"Why can't people just upgrade fuel pumps and increase fuel line (pipe?) pressures to force fuel to flow in to the engine when the intake valves are open instead?"
This is the job of the fuel injector/ECU. It is timed specifically so the proper amount of fuel is injected at the correct time. The ECU is tuned for a specific fuel pressure and can typically compensate if it changes slightly.
If you are asking in terms of modifying for more power...
The ECU and typically compensate for small changes
For larger changes, you will want to retune your ECU. You will not need the same fuel increase throughout the RPM range, and will need to tune increases based on RPM.
Typically, you only need to raise the fuel pressure if you are raising the pressure in the intake by way of turbocharger or supercharger, otherwise, you get injectors that flow more fuel.
"Are there any other alternatives to these methods of fuel injection?"
Carburetor, Mechanical fuel injection (seen in the 50s and 60s), Throttle body injection, multi port fuel injection, direct injection.
A carburetor is mechanical and injects fuel based on air flowing through, plus precise tuning.
Not sure on mechanical fuel injection, heard of it, but never looked into it.
Throttle body injection uses a single fuel injector to inject fuel right before the throttle body.
Multi port injection is the most common since the early/mid 90s. Each cylinder gets its own fuel injector mounted in the intake manifold.
Direct injection is not very common on gasoline engines, but is growing in popularity. Gasoline is injected directly into the cylinder. Haven't looked much into this either. I believe it is both the most efficient and the most complex.