Is this mechanism for the formation of a tosylate correct?
No, $\ce{Cl-}$ will not remove the $\ce{H+}$ because it is not basic enough. The $\mathrm{p}K_\mathrm{a}$ of $\ce{Cl-}$'s conjugate acid ($\ce{HCl}$) is $-7$. The $\mathrm{p}K_\mathrm{a}$ of $\ce{NEt3}$'s conjugate acid ($\ce{HNEt3+}$) is $10.75$ (source). This means that $\ce{HCl}$ almost completely dissociates in solution, and is very unlikely to form $\ce{HCl}$ molecules. $\ce{NEt3}$, on the other hand, is much more basic and will easily pick up the $\ce{H+}$.

Also the mechanism is incorrect. A lot of the arrows are wrong. See here (pyridine serves the same function as $\ce{NEt3}$):