The $F$ statistic is directly related to the total $R^2$ of the regression. So if your covariates, together, explain a lot of the variation of $y$, then you will have a high $F$ statistics.
But the variance of each specific coefficient depends on how the covariates are correlated. To see how that happens, let's consider a linear regression with two covariates $X= [x_1, x_2]$:
$$
y = \hat{\beta}_1x_1 + \hat{\beta}_2x_2 + \hat{\epsilon}
$$
To simplify computation, let the variables be standardized to have unit variance and mean zero. Let $\hat{\beta} = [\beta_1, \beta_2]$. The estimated variance of $\hat{\beta}$ is:
$$
\begin{align}
\widehat{Var(\hat{\beta})} &= (X'X)^{-1}\hat{\sigma} \\
&= Cor(X)^{-1} \frac{SSR}{(n-1)(n-2)}\\
&=Cor(X)^{-1} \frac{(1 - R^2)}{n-2}\\
\end{align}
$$
Where $SSR$ is the sum of squared residuals, $Cor(X)$ is the correlation matrix of $X$ and $R^2$ is the $R^2$ of the whole regression. We are interested in the diagonal elements of this variance, so let's pick the first element of the diagonal, which is $\widehat{Var(\hat{\beta}_1)}$. Since our variables are standardized, the first term of $Cor(X)^{-1}$ is simply $\frac{1}{1 - Cor(x_1, x_2)^2}$. Hence:
$$
\begin{align}
\widehat{Var(\hat{\beta}_1)} &= \frac{(1 - R^2)}{(1 - Cor(x_1, x_2)^2)(n-2)}\\
&= \frac{(1 - R^2)}{(1 - R^2_{x_1 \sim x_2})(n-2)}
\end{align}
$$
Where $R^2_{x_1 \sim x_2}$ is the percentage of the variance of $x_1$ explained by $x_2$. That is, while a large $R^2$ for the whole regression reduces variances overall, if $x_2$ explains a lot of the variance of $x_1$ this can have serious effects on the variance of the coefficient, giving you high p-values.
So your suggested interpretation is not correct. What your results mean is that $\{x_1, x_2\}$ explain a lot of the variation of $y$ together, but since the correlation between them is high, you can't estimate the contribution of each term precisely with your sample size.
(As a side note, it seems you are interpreting p-values as posterior probabilities --- but that would be another whole answer on its own, you should take a look at some of the answers here and here. And if you are trying to use regression for causal inference, you should take a look here.)
