In an arithmetic theory, you don't need an induction axiom to obtain a result $P(n)$ for some specific $n.$ You are correct, you can just use the idea behind induction to take you from $P(0)$ to $P(n)$ step by step, however large $n$ might be, just using simple step-by-step reasoning... no overarching principle needed. Where an induction axiom is necessary is generalizing this to all natural numbers. In logic, there is a big difference between being able to prove, for all $n,$ that "$\phi(n)$ holds," and being able to prove "for all $n,$ $\phi(n)$ holds." In the first case, the quantifier 'for all $n$' is in the metatheory, which doesn't always translate to a quantifier in the formal theory. (This may seem pedantic, but it's actually a very important subtlety that shows up in a lot of important results in logic.)
(Note also there is also a quantification over 'all properties' (more specifically all formulas) that typically happens in the metatheory, at least for first order theories.)
That's how it works in an arithmetical theory like PA, but you asked about the axiom of infinity's bearing here, so we need to see how this looks in set theory. So suppose our formal theory is ZF without the axiom of infinty where we've embedded arithmetic in the usual way where $\mathbf 0\equiv \emptyset$ and $\mathbf S(\mathbf n)\equiv \mathbf n\cup\{\mathbf n\}.$ The property of being a natural number (under this translation) is definable in set theory (it is a natural number if it an ordinal that is not greater than or equal to any limit ordinal). So one can, fairly straightforwardly, translate any first-order statement of arithmetic as a statement about sets that are natural numbers.
Notice here, we are just treating the natural numbers as a class, not necessarily a set. In fact we can prove from ZF minus infinity that any of the (translations of the) axioms of PA, including all the induction axioms, hold in this class. The induction axioms just follow from the transfinite induction theorem schema on the ordinals, which does not involve infinity at all, and just says any subclass of the ordinals has a least element. All that infinity would do is guarantee that $\mathbb N$ is a set and thus not all of ordinals. In fact, it's known that PA and ZF minus infinity (plus the negation of infinity) are, pretty much equivalent theories. Where the axiom of infinity becomes important is for second-order arithmetic (and second order induction) where we need to be able to quantify over arbitrary subsets of $\mathbb N,$ and thus take it seriously as a set.
(Infinity gives some other interesting things things here. This is an instance of the important subtlety I mentioned above. When I said 'any subclass of the ordinals' above, that quantifier was in the metatheory. When we are doing induction up to some ordinal ($\omega$ in this case) we can actually state and prove, in the theory, that every subset of $\omega$ has a least element. On a very related note, while ZF minus infinity can prove, for each instance of PA's induction schema, that $\omega$ (i.e. the class of all ordinals in this case) satisfies the instance, in ZF with infinity we can formalize and prove "$\omega$ satsifies every instance of the induction axiom." This has a lot to do with the fact that ZF can prove the consistency of PA whereas ZF with infinity replaced by its negation, being of the same strength as PA, cannot.)
