The performance of a rocket engine - its specific impulse - is directly proportional to the velocity of exhaust gas (and nothing else!). That velocity is achieved by releasing the combustion products from pressurized combustion chamber (pressurized by continuous production of exhaust gas by burning the fuels) and the higher the pressure the more you can accelerate the exhaust gas - obtain better performance.
To inject fuels into the combustion chamber you need to push them in at pressure higher than present in the chamber. That necessitates plumbing and infrastructure capable of withstanding these pressures - thick-walled, bulky and heavy. If you pressurize the entire tank, the entire tank must be pressure-proofed - made robust enough to withstand the high pressures. That will result either in exceptionally thick, heavy tank, or - practically - a tank that is moderately heavy but only holds a very moderate pressure. That converts to low combustion chamber pressure and poor performance.
The turbopumps are a way around this - the tank must only withstand very modest pressure needed to get the fuel to the pump, and then only the small segment of the infrastructure past the pump must be reinforced, and it can be reinforced by quite a bit (it's small!) providing very high chamber pressure - great engine performance.
Still, pressure-fed rockets aren't all that uncommon; most of early-days rocket engines were pressure fed. It's often more economical to go with a simpler, bigger, low-performance stage of a rocket than to develop something of excellent performance that just costs a lot.
As for pressurizing through gravity and acceleration - 10m of water produces 1 bar of pressure differential in 1g. Liquid oxygen, RP-1, hydrogen, methane etc are all less dense, but let's use water for ballpark numbers and upper bound using some extremes. Saturn V was 111 meters tall. Let's give it a pretty oppressive 6g of acceleration and run the fuel from the very tip to the engine. You're still getting only the very modest 66 bars. You could improve it by pressurization but you're already carrying all this mass on top of a 111m tall tower, the structural overhead will be massive! Meanwhile, SpaceX's Merlin, their workhorse, goes at nearly 100 bar and is rather mediocre performance-wise.