Variance and empirical distribution 
We know that regressing two I(1) time series that are not cointegrated might give rise to spurious results. Indeed, we might have an excess of rejection of the null hypothesis due to the fact that as the sample size increases, the empirical distribution of the t-stat becomes fatter and fatter, and this makes the critical values move to the left and to the right. This implies that if I stay with the critical values of the Student-$t$, I will have a rejection of the null hypothesis as shown in the sketch above.
Is my intuition correct?

Basically, the intuition is mostly correct. It is however questionable if it makes sense to call the quantiles of the distribution of the t-statistic (not so much the estimator, as you compare t-statistics to criticical values of, e.g., the $t_{n-2}$, or, more commonly and based on asymptotic arguments, the $N(0,1)$) "critical values".
Actually, it is known from Phillips (1986) that the t-statistic diverges at rate $T^{1/2}$, so indeed will exceed any finite critical values with probability 1.
A little illustration:

library(RColorBrewer)

t.spurious <- function(n){ 
  y <- cumsum(rnorm(n)) 
  x <- cumsum(rnorm(n)) 
  summary(lm(y~x))$coefficients[2,3] 
} 

n <- c(50, 100, 200, 500, 1000)
t.stats <- lapply(n, function(n) replicate(5000, t.spurious(n)))
 
colors <- brewer.pal(5, "Set1") 
densities <- lapply(t.stats, density)
plot(densities[[1]], lwd=2, col=colors[1])
lines(densities[[2]], lwd=2, col=colors[2])
lines(densities[[3]], lwd=2, col=colors[3])
lines(densities[[4]], lwd=2, col=colors[4])
lines(densities[[5]], lwd=2, col=colors[5])
legend("topleft", legend = n, col=colors, lty = 1, lwd = 2)

