Why does the PLT exist in addition to the GOT, instead of just using the GOT?

The problem is that replacing call printf@PLT with call [printf@GOTPLT] requires that the compiler knows that the function printf exists in a shared library and not a static library (or even in just a plain object file). The linker can change call printf into call printf@PLT, jmp printf into jmp printf@PLT or even mov eax, printf into mov eax, printf@PLT because all it's doing it changing a relocation based on the symbol printf into relocation based on the symbol printf@PLT. The linker can't change call printf into call [printf@GOTPLT] because it doesn't know from the relocation whether it's a CALL or JMP instruction or something else entirely. Without knowing whether it's a CALL instruction or not, it doesn't know whether it should change the opcode from a direct CALL to a indirect CALL.

However even if there was a special relocation type that indicated that the instruction was a CALL, you still have the problem that a direct call instruction is a 5 bytes long but a indirect call instruction is 6 bytes long. The compiler would have to emit code like nop; call printf@CALL to give the linker room to insert the additional byte needed and it would have to do it for all calls to any global function. It would probably end up being a net performance loss because of all the extra and not actually necessary NOP instructions.

Another problem is that on 32-bit x86 targets the PLT entries are relocated at runtime. The indirect jmp [xxx@GOTPLT] instructions in the PLT don't use relative addressing like the direct CALL and JMP instructions, and since the address of xxx@GOTPLT depends on where the image was loaded in memory the instruction needs to be fixed up to use the correct address. By having all these indirect JMP instructions grouped together in one .plt section means that much smaller number of virtual memory pages need to be modified. Each 4K page that's modified can no longer be shared with other processes, when the instructions that need to modified are scattered all over memory it requires that a much larger part the image to be unshared.

Note that this later issue is only a problem with shared libraries and position independent executables on 32-bit x86 targets. Traditional executables can't be relocated, so there's no need to fix the @GOTPLT references, while on 64-bit x86 targets RIP relative addressing is used to access the @GOTPLT entries.

Because of that last point new versions of a GCC (6.1 or later) support the -fno-plt flag. On 64-bit x86 targets this option causes the compiler to generate call printf@GOTPCREL[rip] instructions instead of call printf instructions. However it appears to do this for any call to a function that isn't defined in the same compilation unit. That is any function it doesn't know for sure isn't defined in shared library. That would mean that indirect jumps would also be used for calls to functions defined in other object files or static libraries. On 32-bit x86 targets the -fno-plt option is ignored unless compiling position independent code (-fpic or -fpie) where it results in call printf@GOT[ebx] instructions being emitted. In addition to generating unnecessary indirect jumps, this also has the disadvantage of requiring the allocation of a register for the GOT pointer though most functions would need it allocated anyways.

Finally, Windows is able to do what you suggest by declaring symbols in header files with the "dllimport" attribute, indicating that they exist in DLLs. This way the compiler knows whether or not to generate direct or indirect call instruction when calling the function. The disadvantage of this is that the symbol has to exist in a DLL, so if this attribute used is you can't decide after compilation to link with a static library instead.

Read also Drepper's How to write a shared library paper, it explains that quite well in details (for Linux).

Now I'm wondering why there are two indirections (calling into the PLT and then jumping to an address from the GOT),

First of all there are two calls, but just one indirection (call to PLT stub is direct).

instead of just sparing the PLT and calling the address from the GOT directly.

In case you do not need lazy binding, you can use -fno-plt which bypasses the PLT.

But if you wanted to keep it, you'd need some stub code to see if symbol has been resolved and branch accordingly. Now, to facilitate branch prediction, this stub code has to be duplicated for every called symbol and voila, you re-invented the PLT.