How much rigour is this proof of multivariable chain rule?

This is not rigorous at all. For one thing, you have not even defined most of your notation: what do $\Delta x(t)$, $\delta f_x(x,y)$, and so on mean? Even filling in reasonable guesses for what the notation means, there are serious issues. For instance, if $x(t)$ is a constant function, then it would seem that what you are referring to as $\delta x(t)$ is always $0$, so you cannot divide by it. There is also an issue that the difference $f(x+\Delta x,y+\Delta y)-f(x,y+\Delta y)$ is taken at $y+\Delta y$ instead of at $y$, and so you cannot expect it to be well-approximated using a partial derivative of $f$ at $(x,y)$ unless you know that partial derivative is continuous.

At best, what you have written is a sketch of a proof of the chain rule under significantly stronger hypotheses than you have stated.