I am studying the maximum value of a Brownian Motion (BM) on an interval of time (as explained here between boxes 28 and 40) and I am having an issue aligning intuition with the mathematical result. It is claimed (in box #37) that the probability of the maximum of a standard BM, $X_s$, to exceed the value $y$, denoted as $\mathbb{P}(Y_t \ge y)$ where $Y_t = \text{max}\{X_s : 0 \le s \le t \}$, is equal to the cumulative distribution function (cdf) of a standard Half-Normal distribution with scale parameter $t$, i.e. $\mathbb{P}(Y_t \ge y) = \frac{2}{\sqrt{2 \pi t}} \int_y^\infty e^{-x^2/(2t)} dx$. This integral is the $erf$ function with argument $\frac{x}{\sqrt{2t}}$, and this Half-Normal c.d.f can be equally represented as $erf(\frac{y}{\sqrt{2t}})$.
I have the following intuitions regarding the maximum process, and an issue interpreting this result in that intuitive context.
Intuition #1: For a fixed distance, $y$, as $t$ grows the probability that the value $y$ is reached or exceeded (i.e. $\mathbb{P}(Y_t \ge t)$) will increase. In other words, the longer you wait, the more likely the BM is to reach some fixed distance ($y)$. (In the limiting case, this even leads to the hitting-time property of the standard BM that with probability $1$ the BM visits every value in the state space, $\mathbb{R}$. This is stated in box #33 in the link above). Summary: $y\, -, t \uparrow \implies \mathbb{P}\uparrow$
Intuition #2: As the value $y$ which is of interest to be exceeded approaches further from $0$, the probability that the maximum on the interval will equal to or exceed that value (i.e. $\mathbb{P}(Y_t \ge y)$) will decrease, for a fixed waiting time, $t$. In other words, for a given duration of time, the further away the BM is required to reach, the less likely it is to reach that far. Summary: $y \uparrow, t \, - \implies \mathbb{P} \downarrow$
My problem is that it seems to me that as time, $t$, is increased in the probability definition above, $\mathbb{P}(Y_t \ge y)=erf(\frac{y}{\sqrt{2t}})$, that the resulting probability decreases, and this conflicts with Intuition #1 above. That is, the $erf$ function is sigmoid shaped and in the range of interest (value in $[0,1]$, since it is a probability; $t \in [0,\infty)$ and $y>0$ by assumption) decreasing the value of the input argument decreases the function's value. Since the variable $t$ is in the denominator in $erf(\frac{y}{\sqrt{2t}})$, the resulting effect is that increasing $t$ decreases $\mathbb{P}(Y_t \ge y)$, which conflicts with intuition.
Intuition #2 also seems in conflict since $y$ is in the numerator and thus has an increasing effect on the monotonic function $erf$ as it is increased.
Any advice for clarifying my confusion would be appreciated.