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:
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/MoHoj.jpg)
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)