Problems understanding proof of smn theorem using Church-Turing thesis
In books on computability theory it is common for the text to skip details on how a particular machine is to be constructed. The author of the computability book will mumble something about the Turing-Church thesis somewhere in the beginning. This is to be read as "you will have to do the missing parts yourself, or equip yourself with the same sense of inner feeling about computation as I did". Often the author will give you hints on how to construct a machine, and call them "pseudo-code", "effective procedure", "idea", or some such. The Church-Turing thesis is the social convention that such descriptions of machines suffice. (Of course, the social convention is not arbitrary but rather based on many years of experience on what is and is not computable.)

I am not saying that this is a bad idea, I am just telling you honestly what is going on. Books on analysis skip $\epsilon\delta$-proofs, books on category theory don't verify every naturality square, books on topology rely on geometric intuitions of the readers, etc.

There are standard tricks on how to read computability theory. If the book says "there exists $X$", then that usually means "$X$ can be computed from whatever our current parameters are". If they want to emphasize that whatever they are doing can be computed as a function of a parameter $p$, they will say "whatever we are doing, uniformly in $p$". And obviously, if they say "non-uniformly in $p$", they are emphasizing that whatever they are doing is not computable as a function of $p$.

In Barry's text, he could have emphasized the computability bit by writing "there exists an index $e_{x_0}$, uniformly in $x_0$, such that ..." But since this is an introductory text, sticking in the word "uniformly" will just confuse a good part of the readership. So what are we supposed to do? We certainly do not want to write out detailed constructions of machines, because then students will end up thinking that's what computability theory is about. It isn't. Computability theory is about contemplating what machines we could construct if we wanted to, but we don't. As usual, the best path to wisdom is to pass through a phase of confusion.