Black--Scholes hedging argument

I'm trying to understand the standard hedging argument to derive the Black--Scholes PDE. There's one aspect of the derivation which I can't get passed and I'd be very grateful for some clarification here.

We make the standard assumptions: underlying follows a geometric BM, i.e. $\text{d} S_t = \mu S_t \text{d}t + \sigma S_t \text{d}W_t$ and we let $V(S_t,t)$ be an option written on this security so that by Ito's Lemma, we have:

$\text{d} V_t = \left(\frac{\partial V}{\partial t} + \mu S_t \frac{\partial V}{\partial S} + \frac{1}{2} \sigma^2 S^2_t \frac{\partial^2 V}{\partial S^2}\right) \text{d}t + \left(\sigma S_t \frac{\partial V}{\partial S}\right) \text{d}W_t$

All good so far. We then set-up a portfolio consisting of a single option and some amount $\Delta(S_t,t)$ of the underlying whose value is given by $\Pi_t = V_t + \Delta(S_t,t) S_t$. It is then assumed that this portfolio be self-financing, i.e. that there are no additional inflows or outflows of cash, formalised as $\text{d} \Pi_t = \text{d} V_t + \Delta(S_t,t) \text{d} S_t$. We then choose $\Delta(S_t,t) = - \frac{\partial V}{\partial S}$ and then by substituting this into the above formula and using self-financing, we quickly obtain a riskless portfolio and the Black--Scholes PDE follows from there.

The point that confuses me is that it is not clear to me that we can definitely construct a portfolio satisfying $\Pi_t = V_t - \frac{\partial V}{\partial S} S_t$ which also must satisfy $\text{d} \Pi_t = \text{d} V_t - \frac{\partial V}{\partial S} \text{d} S_t$. Every text I can find simply assumes that you can and proceeds as above. However, the existence of such a portfolio implies that $\frac{\partial V}{\partial S} S_t = \int_0^t \frac{\partial V}{\partial S}(S_u,u) \text{d}S_u$ and I don't see that this is necessarily true, that is to say, I don't see how such a portfolio can exist? What am I missing?

The interesting thing is that I have followed a different derivation of Black--Scholes where the portfolio instead consists of some amount of the underlying and some amount of a risk-free instrument $B_t$, i.e. $\Pi_t = \alpha(S_t,t) S_t + \beta(S_t,t) B_t$. Here we choose $\alpha(S_t,t) = \frac{\partial V}{\partial S}$ as before to remove the risk but this time the freedom in $\beta(S_t,t)$ seems to mean that we can ensure the portfolio is self-financing and therefore I am happy with this version of the argument.

I would really like to understand both arguments and where it is that I am going wrong in the first so any help is gratefully appreciated!