Why would you use the ternary operator without assigning a value for the "true" condition (x = x ?: 1)

This is permitted in GNU as an obscure extension to C

5.7 Conditionals with Omitted Operands

The middle operand in a conditional expression may be omitted. Then if the first operand is nonzero, its value is the value of the conditional expression.

Therefore, the expression

 x ? : y

has the value of x if that is nonzero; otherwise, the value of y.

This example is perfectly equivalent to

 x ? x : y

In this simple case, the ability to omit the middle operand is not especially useful. When it becomes useful is when the first operand does, or may (if it is a macro argument), contain a side effect. Then repeating the operand in the middle would perform the side effect twice. Omitting the middle operand uses the value already computed without the undesirable effects of recomputing it.

As you can probably guess, avoiding this is recommended for readability and portability reasons. I'm honestly surprised to see such a grammar-incompatible extension to C.

This is a GCC extension that means "if the condition is true, use it, else use this other value", so

machine->max_cpus = machine->max_cpus ?: 1;

is shorthand for

machine->max_cpus = machine->max_cpus ? machine->max_cpus : 1;

although if the conditional has side-effects, it will only be run once

Using gcc's -pedantic flag, it does say

foo.c:5: warning: ISO C forbids omitting the middle term of a ?: expression