black scholes - Why a self-financing replicating portfolio should always exist?

According to my understanding the derivation of the Black-Scholes PDE is based on the assumption that the price of the option should change in time in such a way that it should be possible to construct such a self-financing portfolio whose price replicates the price of the option (within a very small time interval). And my question is: Why do we assume that the price of the option has this property?

I will explain myself in more details. First, we assume that the price of a call option $C$ depends on the price of the underlying stock $S$ and time $t$. Then we use the Ito's lemma to get the following expression:

$d C = (\frac{\partial C}{\partial t} + S\mu\frac{\partial C}{\partial S} + \frac{1}{2}S^2 \sigma^2 \frac{\partial^2C}{\partial S^2}) dt + \sigma S \frac{\partial C}{\partial S} dW$ (1) ,

where $\mu$ and $\sigma$ are parameters which determine the time evolution of the stock price:

$dS = S(\mu dt + \sigma dW)$ (2)

Now we construct a self-financing portfolio which consist of $\omega_s$ shares of the underlying stock and $\omega_b$ shares of a bond. Since the portfolio is self financing, its price $P$ should change in this way:

$dP(t) = \omega_s dS(t) + \omega_b dB(t)$. (3)

Now we require that $P=C$ and $dP = dC$. It means that we want to find such $\omega_s$ and $\omega_b$ that the portfolio has the same price that the option and its change in price has the same value as the change in price of the option. OK. Why not? If we want to have such a portfolio, we can do it. The special requirements to its price and change of its price should fix its content (i.e. the requirement should fix the portion of the stock and bond in the portfolio ($\omega_s$ and $\omega_b$)).

If we substitute (2) in (3), and make use of the fact that $dB = rBdt$ we will get:

$\frac{\partial C}{\partial t} + S\mu\frac{\partial C}{\partial S} + \frac{1}{2}S^2 \sigma^2 \frac{\partial^2C}{\partial S^2} = \omega_s S \mu + \omega_b r B$ (4)

and

$\sigma S \frac{\partial C}{\partial S} = \omega_s S \sigma$ (5)

From last equation we can determine $\omega_s$:

$\omega_s = \frac{\partial C}{\partial S}$ (6)

So, we know the portion of the stock in the portfolio. Since we also know the price of the portfolio (it is equal to the price of the option), we can also determine the portion of the bond in the portfolio ($\omega_b$).

Now, if we substitute the found $\omega_s$ and $\omega_b$ into the (4) we will get an expression which binds $\frac{\partial C}{\partial t}$, $\frac{\partial C}{\partial S}$, and $\frac{\partial^2 C}{\partial S^2}$:

$\frac{\partial C}{\partial t} + rS \frac{\partial C}{\partial S} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} = rC$

This is nothing but the Black-Scholes PDE.

What I do not understand is what requirement binds the derivatives of $C$ over $S$ and $t$.

In other words, we apply certain requirements (restrictions) to our portfolio (it should follow the price of the option). As a consequence, we restrict the content of our portfolio (we fix $\omega_s$ and $\omega_b$). But we do not apply any requirements to the price of the option. Well, we say that it should be a function of the $S$ and $t$. As a consequence, we got the equation (1). But from that we will not get any relation between the derivatives of $C$. We, also constructed a replicating portfolio, but why its existence should restrict the evolution of the price of the option?

It looks to me that the requirement that I am missing is the following:

The price of the option should depend on $S$ and $t$ in such a way that it should be possible to create a self-financing portfolio which replicates the price of the option.

Am I right? Do we have this requirement? And do we get the Black-Scholes PDE from this requirement? If it is the case, can anybody, please, explain where this requirement comes from.