I don't think you want to interpret the filtration $\mathcal{F}_t$ as the events that can be assigned a probability at the time $t$. It is more natural to interpret it as the information present at time $t$. Let me explain:
The usual setup is that we have a probability space $(\Omega,\mathcal{F},P)$, i.e. $\Omega$ is a non-empty set, $\mathcal{F}$ is a sigma algegra on $\Omega$ and $P$ is a probability measure on $\mathcal{F}$. Here we have already said that the sets/events in $\mathcal{F}$ are the sets/events we can assign a probability, and so this does not vary over time.
Now we can equip our probability space with a filtration $(\mathcal{F}_t)_{t\geq 0}$, that is $\mathcal{F}_t$ are sigma algebras with $\mathcal{F}_t\subseteq \mathcal{F}$ for all $t$ such that
$$
\mathcal{F}_s\subseteq \mathcal{F}_t\quad\text{whenever }s\leq t.\tag{1}
$$
So you can see why we don't want to think of $\mathcal{F}_t$ as the sets/events that we can assign a probability because this is at any time given by the whole of $\mathcal{F}$.
Instead, think of $\mathcal{F}_t$ as the information present at time $t$. Then $(1)$ tells us that we are not getting dumber with time which is a reasonable assumption. If $X:\Omega\to\mathbb{R}$ is a random variable that is $\mathcal{F}_t$-measurable, then it means that we can determine $X$ based on the information available at time $t$ because
$$
X^{-1}(B)=\{X\in B\}\in\mathcal{F}_t,\quad \text{for all Borel sets }B.
$$
Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space equipped with a filtration $(\mathcal F_t)_{t\in\mathbb R_+}$.
For all $t\in\mathbb R_+$, let $Y_t:(\Omega,\mathcal F)\to(\Omega,\mathcal F_t)$ be defined for all $\omega\in\Omega$ by $Y_t(\omega)=\omega$. Then we clearly have $\sigma(Y_s,s\le t)=\mathcal F_t$, hence $(\mathcal F_t)_{t\in\mathbb R_+}$ is the natural filtration of the stochastic process $Y:(\Omega,\mathcal F)\to(\Omega^{\mathbb R_+},\bigotimes_{t\in\mathbb R_+}\mathcal F_t),\omega\mapsto(\omega)_{t\in\mathbb R_+}$.
So for any filtration $(\mathcal F_t)_{\mathbb R_+}$, there exists a measurable space $(E,\mathcal E)$ and a stochastic process $Y:\Omega\to E$ such that for all $t\in\mathbb R_+$, $\mathcal F_t=\sigma(Y_s,s\le t)$.
A much more interesting question would be to know whether this is true when $(E,\mathcal E)$ is fixed. For instance, for any filtration $(\mathcal F_t)_{t\in\mathbb R_+}$, does there exists a real-valued process $(Y_t:\Omega\to(\mathbb R,\mathcal B(\mathbb R))_{t\in\mathbb R_+}$ such that $(\mathcal F_t)_{t\in\mathbb R_+}$ is the natural filtration of $Y$? The answer is no, as shown is this post Is every sigma-algebra generated by some random variable?.
In many pratical scenarios like quantitative finance, as you said, you are very likely to deal with the natural filtration of an observable process $Y$ and you are interested in a different process $X$. But when $X$ is a one-to-one and onto function of $Y$, then the natural filtrations of $X$ and $Y$ are the same.
Best Answer
You know the probability of every event in every $\mathcal{F}_t$, but the idea is that at time $t$ you know specifically which event you are in. For an easy example, you can think of flipping two fair coins. The sample space is $\Omega = \{ HH, HT, TH, TT\}$, and the terminal $\sigma$-algebra is $\mathcal{F} = 2^\Omega$. Then $\mathcal F_n$ will be what you know from observing the first $n$ coin tosses. $\mathcal F_0 = \{\emptyset, \Omega\}$ is just the trivial $\sigma$-algebra because you don't have any information (except whether or not the coins were actually tossed), so you can just say the probability that you end up in any particular event in $\mathcal F$. Now $\mathcal F_1 = \{\emptyset, \Omega, \{HH, HT\}, \{TH, TT \} \}$ because you will know exactly which of those events you are in after observing the first toss; e.g. if it was $H$, you're in $\{HH, HT\}$ (and $\Omega$). And $\mathcal F_2 = \mathcal F = 2^\Omega$ because after observing both coin tosses you know exactly what the outcome is, so you can say for sure whether or not you're in any particular subset of $\Omega$.