While reading about minimal instruction set computer I found out that one needs at least (for example) the ability to increment or decrement the value stored in register, a test for zero and a jump.

A machine with such instruction set would be Turing-complete, correct? Is this called "Counter machine"?

But then I found out about the so-called "Random-access machine" which adds "indirect addressing", meaning I can access a register not just by explicitly specifying memory address or label but I can also fetch an address out of an address. This should be useful for enabling mechanisms akin to "pointers" in C-like languages, for example.

My question is, why there is the need for "indirect addressing"? Since I presume both machines (without and with indirect addressing) are Turing complete, is it "merely" for convenience? Or are some programs which cannot by expressed without indirection?

PS: I'm complete noob in this area so please bear with me: if I was to implement memcpy, i.e. to transfer the values of bunch of registers to other bunch of registers, how would I do it without indirection? I mean, I can write copy 1 word from x to y, copy 2 words from x to y, ... but in general, how do I say copy n words from x to y without it?

Edit:

For example, imagine I receive zero-terminated string via network, say "1 2 3 0" which I would like to store to some of my infinitely-many registers. Receiving "0" halts the program. I have a instruction "take r" which copies one word from the input to register "r". No indirection means (?) that "r" has to be direct constant number. How do I then move to "n-th" register? In other words, I don't understand how do I move back and forth among the registers, when the only mechanism I have is to read/set register 1, 2, 3, ...?

Idea: Addresses are offsets. If you want the $n$th cell, move a marker $n$ steps from the beginning. For that, you maintain a counter at a fixed location (determined by the programmer/compiler). After that, perform your operation on the first marked cell. Remove the marking when you are done.