I cannot find a good reference for the proof that the ring of integers in a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$. The proof I usually find does an induction on the number of prime factors of $n$, coupled with a lengthy and somewhat computational proof in the case where $n$ is the power of a prime.
Do you know a quicker and possibly more conceptual approach?
The ring of integers $R_n$ of ${\mathbf Q}(\zeta_n)$ contains ${\mathbf Z}[\zeta_n]$ as a subring with finite index. To show the containment of rings is an equality, it suffices to show the inclusion ${\mathbf Z}[\zeta_n] \rightarrow R_n$ becomes an isomorphism after tensoring with ${\mathbf Z}_p$ for all $p$, and this basically boils down to showing the ring of integers of ${\mathbf Q}_p(\zeta_n)$ is ${\mathbf Z}_p[\zeta_n]$ for all $p$. Now you have to know something about how to compute rings of integers in unramified and totally ramified extensions of local fields. I'm leaving off some details here, admittedly, and since I used the terrible word "compute" maybe this isn't an answer you are looking for.
You didn't tell us whether you were okay with the inductive argument but disliked (apparently) the prime-power case or you were unhappy with both aspects.
In case of a prime power it is not so hard. Denote $R$ the ring of integers of ${\mathbf Q}(\zeta_n)$. First note that $X^n -1$ is separable mod $p$ when $p$ does not divide $n$. This implies $R [1/p] = {\mathbf Z}[\zeta_n, 1/p]$. To check that $R = {\mathbf Z}[\zeta_n]$ it then suffices to show that the local rings of ${\mathbf Z}[\zeta_n]$ at all primes above $p$ are DVRs. But you have the explicit element $\lambda = 1 - \zeta$ that you defined above and its norm equals $p$. This implies that there is only one prime above $p$ and that this ideal is generated by $\lambda$. In particular the local ring at this prime is regular and hence a DVR.
I don't know if this is what you're looking for, but I like the proof in Neukirch's Algebraic Number Theory Section I.10. Conceptually the idea (for $n$ a prime power) is that if the discriminant is a prime power and there is an element $\lambda$ whose norm is that prime, then $\mathbf{Z}[\lambda]$ should be the ring of integers. Here, $\lambda = 1- \zeta$ so $\mathbf{Z}[\lambda] = \mathbf{Z}[\zeta]$.
Then for general $n$ you use the fact that $\mathbf{Q}(\zeta_n)$ is the compositum of $\mathbf{Q}(\zeta_{\ell^r})$, and $\zeta_{\ell^r}$ is a power of $\zeta_n$.
I recently found a very short proof of the prime power case in the incredibly enlightening notes of Peter Stevenhagen: Number rings.
The proof follows almost entirely from the Dedekind–Kummer theorem for orders $\mathbb{Z}[\alpha]$ (Theorem 3.1 in the notes). The theorem states that if $f(x) \in \mathbb{Z}[x]$ is the minimal polynomial of $\alpha$ and $\overline{f} = \prod \overline{g}_i^{e_i}$ mod $p$, then the primes $\mathfrak{p}$ above $p$ in $\mathbb{Z}[\alpha]$ correspond exactly to the $\overline{g_i}$. Moreover, $\mathfrak{p}$ is 'singular' iff $e_i > 1$ and also $p^2$ divides the remainder $r$ in $f = ag_i + r$. I have left out some details which are in Stevenhagen's notes.
To apply this to $\mathbb{Z}[\zeta_{p^k}]$, write $\Phi_{p^k}(x) = \frac{x^{p^k} - 1}{x^{p^{k-1}}-1}$. Mod $p$, this is just $\frac{(x-1)^{p^k}}{(x-1)^{p^{k-1}}} = (x-1)^{p^k - p^{k-1}}$ which shows $\mathfrak{p} = (\zeta - 1, p)$ is the only prime above $p$ and also since the remainder of $\Phi_{p^k}$ divided by $x-1$ is just $\Phi_{p^k}(1) = p$ (which is not divisible by $p^2$), it follows that $\mathfrak{p}$ is 'regular'. It is easy to check $\Phi_{p^k}$ and its derivative are coprime mod $q$ for all other primes $q$, implying all the $e_i$'s are 1. Thus, all the primes above $q$ are 'regular' as well (and unramified). All primes being regular implies the ring is Dedekind, so $\mathbb{Z}[\zeta]$ must be the ring of integers. This is covered in more detail in Theorem 3.12 in the notes.
This argument still works prime-by-prime but avoids having to localize (of course, the localization is hidden under the hood in the proof of Dedekind–Kummer). Moreover, this approach is just a specific application of the very general framework of applying Dedekind–Kummer to try to determine the ring of integers containing some $\mathbb{Z}[\alpha]$ and the same exact type of analysis will quickly determine the ring of integers of $\mathbb{Q}[\sqrt{d}]$ and $\mathbb{Q}[\sqrt[3]{d}]$, just to name a few.
