I write $m$ instead of $n$ for the number of time points, because $n$ is already used for the number of random processes.
Since each $X^{(i)}$ is a Gaussian process, the random vector $(X_{t_1}^{(i)},\dots X_{t_m}^{(i)})$ has a multivariate normal distribution for each $i=1,\dots,n$. Putting $n$ independent vectors of this kind together makes a normally distributed vector of length $mn$: $(X_{t_1}^{(1)},\dots X_{t_m}^{(1)}, X_{t_1}^{(2)},\dots X_{t_m}^{(2)}, \dots X_{t_1}^{(n)},\dots X_{t_m}^{(n)})$ Pushforward by a linear map that sums the entries corresponding to the same time is also a normal distribution. Thus, $(Z_{t_1} ,\dots Z_{t_m} )$ is normally distributed.
Related: Will normal random variables form a normal random vector?