A process $(X_{t})_{t \in T}$ is $(\mathcal{F}_{t})_{t \in T}$-adpated if for every $t$, $X_{t}$ is $\mathcal{F}_{t}$-measurable.
But since the variable $X_{t}$ is interpreted as the state of process at time $t$ and $\mathcal{F}_{t}$ is interpreted as the information know at time $t$, why we don't require $\mathcal{F}_{s} \subset \sigma(X_{t})$ for $s < t$ ?
On
Given $\{\mathcal{F}_n\}_{n\geq 0}$, a process $\{X_n\}_{n \geq 0}$ is adapted if each $X_n$ is $\mathcal{F}_n$-measurable. On the other hand, for predictable process, the random variables are measurable with respect to slightly smaller $\sigma$-algebras. Indeed, given $\{\mathcal{F}_n\}_{n\geq 0}$, a process $\{Y_n\}_{n \geq 0}$ is predictable if each $Y_n$ is $\mathcal{F}_{n-1}$-measurable. i.e., at each time step $n$, one can predict the next value $Y_{n+1}$ of the process using only the information available at time $n$, that is, the $\sigma$-algebra $\mathcal{F}_n$.
Analogy with gambling: An adapted process $X_n$ represents the cumulative gain or loss after $n$ turns, while a predictable process represents a betting strategy. It stands to reason that your betting strategy at the $n$-th step can depend only on the outcome of the previous $n-1$ steps, but not on the $n$-th step itself.
When you start learning martingale theory, the filtrations that you see are almost always just the natural filtration $\mathcal{F}_t = \sigma(X_s : s \leq t)$, and in this case it's clear that $X$ is adapted to $\mathcal{F}$.
Indeed, most filtrations are defined as a natural filtration for some process. However, we might then construct new processes from our original one without wanting to change the underlying filtration. A classic example would be the maximum process $$ M_t = \sup_{s\leq t} X_s, $$ which is adapted to $\mathcal{F}$, but in which $\mathcal{F}_t$ contains strictly more information that $\sigma(M_s : s \leq t)$.
Also, it is useful to make statements comparing processes like "the maximum of two submartingales is a submartingale." In order to even make sense of this, the two processes must be submartingales with respect to the same filtration even though they will most likely have different natural filtrations.