Derivation of y(t)=H(s)x(t)
Initially, you're pretending that '\$s\$' is a run of the mill, real valued parameter, and then later in the analysis you say, oh no! it's really the Laplace operator. This means that your very first equation is mixing the time domain and the Laplace domain in the same expression, \$e^{st}\$, which is not valid.
Your analysis is not an accurate version of the youtube video, which actually shows that convolution in the time domain is equivalent to multiplication in the Laplace domain.