Lambda Calculus inductive substitution definition
"If one deletes $(\mathrm{d})$ and allows bound variables to be substituted..."


The problem here is that substitution of a bound variable makes little sense, since one substitutes a variable (let's forget for a moment it should be free) with a term, which may very well be an abstraction or application, but not a variable.

Let me illustrate this by example. What if we are trying to substitute a bound variable with a term, which is not a variable? Like this:
$$[(y\; y)\ / \ x]\ \lambda x. x.$$
What should we do?


Produce $\lambda (y\; y). (y\; y)$? It is not a well-formed term, so the answer is no.
Another choice would be to leave the binding instance intact and make $\lambda x. (y\; y)$ a result. Let's not take into account if it makes sense, at least it is syntactically correct. This variant causes us major troubles too: it will mess up scopes and render $\beta$-reduction useless. And our end goal is to define $\beta$-reduction via substitution.



  $(\mathrm{g})$ prevents a bound variable from changing to a free one. It does this by first substituting a bound variable.


It does not use substitution, it uses renaming, which is a different notion.


  "...is $(\mathrm{g})$ strong enough to handle it without messing up everything?"


$(\mathrm{g})$ does not tell us anything about substitution of bound variables, since it has the condition $x \in \mathrm{FV}(P)$