In the prime number theorem, the main difficulty is that it is hard to eliminate the case where the limit does not exist and instead oscillates between some
values, i.e., a lot of effort is needed in justifying your first statement $$\displaystyle \lim_{x \rightarrow \infty} \frac{\psi(x)}{x} =c$$ You can get bounds for $A = \limsup_{x \to \infty} \dfrac{\psi(x)}{x}$ and $a = \liminf_{x \to \infty} \dfrac{\psi(x)}{x}$. In fact, Selberg proved that (before his elementary proof with Erdos) that $A+a = 2$. The crux in the case of PNT is showing that $A = a$.
Once you prove that this limit exists, then it is relatively easy to get its value to be $1$.
EDIT
A better way to write out what you have written would be as follows:
First prove that
\begin{align*}
\sum_{d \leq N} \dfrac{\Lambda(d)}d &= \log N + \mathcal{O}(1)
\end{align*}
The proof for this goes as follows.
We have that $\log(N!) = N \log N + \mathcal{O}(N)$. Also,
\begin{align}
\log(N!) & = \sum_{d \leq N} \Lambda(d) \left \lfloor \dfrac{N}d \right \rfloor = \sum_{d \leq N} \Lambda(d) \left( \dfrac{N}d + \mathcal{O}(1)\right)\\
& = N \sum_{d \leq N} \dfrac{\Lambda(d)}d + \mathcal{O} \left(\sum_{d \leq N} \Lambda(d)\right) = N \sum_{d \leq N} \dfrac{\Lambda(d)}d + \mathcal{O} \left(N\right)
\end{align}
Hence, $$\sum_{d \leq N} \dfrac{\Lambda(d)}d = \log N + \mathcal{O}(1)$$
Now use Abel summation technique or by writing $\sum_{d \leq N} \dfrac{\Lambda(d)}d$ as $\displaystyle \int_{2^-}^x \dfrac{\psi(t)}{t}dt$ and performing integration by parts to conclude that to conclude that if $\lim_{x \to \infty} \dfrac{\psi(x)}x = c$ exists, then $c=1$.
Let's begin by presenting a simple example of representing a function with a series. Let $f(z)=\frac{1}{1-z}$ for $z\ne 1$. Recall that $f(z)$ can be represented by the geometric series
$$f(z)=\sum_{n=0}^\infty z^n \tag 1$$
for $|z|<1$. So, although $f(z)$ exists for all $z\ne 1$, its representation as given in $(1)$ is valid only when $|z|<1$.
We can also represent $f(z)$ by the series
$$f(z)=-\sum_{n=1}^\infty \left(\frac{1}{z}\right)^n \tag 2$$
for $|z|>1$. So, we have two representations for the same function that are valid in distinct regions of the complex $z$-plane.
Now, suppose that we represent the function denoted $\zeta(s)$ by the series
$$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}$$
for $\text{Re}(s)>1$. We can easily extend the definition by writing
$$\underbrace{\sum_{n=1}^{\infty}\frac{1}{n^s}}_{\zeta(s)}=\underbrace{2\sum_{n=1}^{\infty}\frac{1}{(2n)^s}}_{2^{1-s}\zeta(s)} -\sum_{n=1}^{\infty} \frac{(-1)^n}{n^s}\tag 3$$
for $\text{Re}(s)>1$. Upon rearranging $(3)$ we find
$$\zeta(s)=(1-2^{1-s})^{-1}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^s}\tag 4$$
But notice that the series on the right-hand side of $(4)$ converges for $\text{Re}(s)>0$. So, we have just developed another representation for $\zeta(s)$ that is valid in a larger region of the complex $s$-plane.
There are other series representations of the Riemann Zeta function, such as its Laurent series,
$$\zeta(s)=\frac1{s-1}+\sum_{n=1}^\infty \frac{(-1)^n\gamma_n}{n!}(s-1)^n$$
which converges for all $s\ne 1$.
And there are integral representation of $\zeta(s)$ such as
$$\zeta(s)=\frac1{\Gamma(s)}\int_0^\infty \frac{x^{s-1}}{e^x-1}\,dx$$
which converges for $\text{Re}(s)>1$ and (see 25.5.11 this reference)
$$\zeta(s)=\frac12 +\frac{1}{s-1}-2 \int_0^\infty \frac{\sin(s\arctan(x))}{(1+x^2)^{s/2}(e^{2\pi x}+1)}\,dx$$
which is valid for all $s\ne 1$.
Best Answer
As KCd explains in a comment, the proof of the PNT in Hardy's time seemed to be intimately connected to the complex analytic theory of the $\zeta$-function. In fact, it was known to be equivalent to the statement that $\zeta(s)$ has no zeroes on the half-plane $\Re s \geq 1$. Although this equivalence may seem strange to someone unfamiliar with the subject, it is in fact more or less straightforward.
In other words, the equivalence of PNT and the zero-freeness of $\zeta(s)$ in the region $\Re s \geq 1$ does not lie as deep as the fact that these results are actually true.
The possibility that one could then prove PNT in a direct way, avoiding complex analysis, seemed unlikely to Hardy, since then one would also be giving a proof, avoiding complex analysis, of the fact that the complex analytic function $\zeta(s)$ has no zero in the region $\Re s \geq 1$, which would be a peculiar state of affairs.
What added to the air of mystery surrounding the idea of an elementary proof was the possibility of accessing the Riemann hypothesis this way. After all, if one could prove in an elementary way that $\zeta(s)$ was zero free when $\Re s \geq 1$, perhaps the insights gained that way might lead to a proof that $\zeta(s)$ is zero free in the region $\Re s > 1/2$ (i.e. the Riemann hypothesis), a statement which had resisted (and continues to resist) attack from the complex analytic perspective.
In fact, when the elementary proof of PNT was finally found, it didn't have the ramifications that Hardy anticipated (as KCd pointed out in his comment).
For a modern perspective on the elementary proof, and a comparison with the complex analytic proof, I strongly recommend Terry Tao's exposition of the prime number theorem. In this exposition, Tao is not really concerned with elementary vs. complex analytic techniques as such, but rather with explaining what the mathematical content of the two arguments is, in a way which makes it easy to compare them. Studying Tao's article should help you develop a deeper sense of the mathematical content of the two arguments, and their similarities and differences.
As Tao's note explains, both arguments involve a kind of "harmonic analysis" of the primes, but the elementary proof works primarily in "physical space" (i.e. one works directly with the prime counting function and its relatives), while the complex analytic proof works much more in "Fourier space" (i.e. one works much more with the Fourier transforms of the prime counting function and its relatives).
My understanding (derived in part from Tao's note) is that Bombieri's sieve is (at least in part) an outgrowth of the elementary proof, and it is in sieve methods that one can look to find modern versions of the type of arguments that appear in the elementary proof. (As one example, see Kevin Ford's paper On Bombieri's asymptotic sieve, which in its first two pages includes a discussion of the relationship between certain sieving problems and the elementary proof.) But I should note that modern analytic number theorists don't pursue sieve methods out of some desire for having "elementary" proofs. Rather, some results can be proved by $\zeta$- or $L$-function methods, and others by sieving methods; each has their strengths and weaknesses. They can be combined, or played off, one against the other. (The Bombieri--Vinogradov theorem is an example of a result proved by sieve methods which, as far as I understand, is stronger than what could be proved by current $L$-function methods; indeed, it an averaged form of the Generalized Riemann Hypothesis.)
To see how this mixing of methods is possible, I again recommend Tao's note. Looking at it should give you a sense of how, in modern number theory, the methods of the two proofs of PNT (elementary and complex analytic) are not living in different, unrelated worlds, but are just two different, but related, methods for approaching the "harmonic analysis of the primes".