One way it helps me to think about this is in terms of a coin toss. The process of an $n$-step discrete random walk in 1d is the same flipping $2n$ couns, getting $h$ heads, and taking $2*(h-n)$. If I flip a fair coin 10 million times, how likely is it I'll get exactly 5 million heads? Very unlikely, right? And if I flip 10 billion coins, it's even less likely I'll get exactly 5 billion heads. So, the coins make it obvious to me at least that the distribution gets more spread out in terms of the actual number of heads you get, so the expected RMS gets bigger, and indeed as big as you want, which isn't maybe that hard to guess since it's intuitively going up.
For the expected proportion though, the coins make it clear too. Because, if I flip 10 million fair coins, even though it's very unlikely I get exactly 5 million heads, it's also pretty clearly unlikely that 51% of them will come up heads. And if I flip 10 billion coins it's even less likely. So, intuitively, the expected percent difference is going to zero, and that's again equivalent to the expected variance in the proportion of steps in either direction in your random walk going to zero.
By the way, if you're familiar with the Demoivre-Laplace theorem, this can pretty well be rigorized in a somewhat more intuitive way, since the distributions are widening, so the expected RMS is increasing, but on taking proportions we contract the number line so that they're shortening for proportions, since your contracting by more and more, which is pretty much a totally rigorizable argument.