In principle you could implement a random button yourself using a link rotator. What's a link rotator? It's basically a link that can take you to one out of several destinations, with each destination assigned a certain probability. Make it equiprobable and you've got your own random button.

Unfortunately, even though there are some web services that offer link rotators, they are, as far as I know, either paid or they limit the number of clicks they allow.

But I agree 100%. Medium should absolutely implement a random button and maybe also a tag cloud. Currently it's just way too much work to dig into the archives of writers.