Relative basicity of fluorine and oxygen in silyl ether deprotection Why does the $\ce{O}$ take the $\ce{H+}$ instead of the $\ce{F}$?

What I took into consideration:


$\ce{O}$ pulls more $e^-$ from atoms near it: $\ce{O}$ is more basic
$\ce{F}$ has one more $e^-$ pair than $\ce{O}$: $\ce{F}$ is more basic
$\ce{F}$ is more electronegative than $\ce{O}$   : $\ce{O}$ is more basic(?)
$\ce{O}$ has more substituent group making it sterically hindered: doesn't matter(?)        


Is it okay to think this way? What's the most important factor?
Because oxygen has a lower electronegativity than fluorine, oxygen is less stable with a negative charge (and more stable with a positive charge). This is why $\ce{OH^-}$ is more basic than $\ce{F^-}$ and why $\ce{HF}$ in water is an acid (hydrofluoric acid), while $\ce{H2O}$ in water is neutral.