The biggest problem is that reddit has a personality. It doesn't just have a tendency or a leniency imposed by the fundamental platform constraints. You go to Twitter when you want short updates or broadcasts from any type of people. Everyone can use it and feel comfortable on Twitter. If someone doesn't want to hear what Alexis Ohanian or people like him think, they can simply unfollow (or not follow in the first place). But it doesn't work that way on reddit.
reddit has a very monotonous, uniform identity and archetype. The admins obviously know this and it's been a big part of reddit's success up to this point, but now, if reddit wants to continue to grow and influence other circles, it has to sanitize from any type of specific personality or affiliation.
It's more than just allowing ostracized subs as a matter of free speech. Yes, subs on reddit can exist for almost anything, but they're starved, brigraded, and slanted. The audience is very hostile and actively tries to intimidate and scare away people who don't match their intended philosophy. It's not like Twitter where you may upset a group and have to deal with some upset tweets. On reddit, if you earn the ire or attention of most redditors, you're going to be in deep shit and your community will be inoperable.
Show reddit to your mom and she'll be appalled by the front page, she'll be out of there by the time she gets to the second link and see that it's anti-Republican or has the "f-word" in it. That's all fine for what reddit has been so far (a place for left-leaning counter-culture nerds to talk about left-leaning counter-culture nerd stuff), but it's not fine for what reddit wants to be (a universal platform for communities of all types and affiliations).
Interestingly this conflicts with your larger point to some extent. Reddit doesn't just need to introduce more subreddits to users, it needs to introduce different users to the platform. The convention on reddit right now seems to be that rational conservatives or religious people or others don't exist, and you'll see stuff that's openly hostile to these groups all over the front page, but in reality, a huge amount of people are conservative, religious, or something else that reddit in general openly loathes.
reddit has to figure out how to make a community that is gentle and welcoming to these people, not hostile or intimidating, and I don't mean that as a general "your UI is not good enough", but I mean it in the very real sense that most people will be very upset at the hostility they encounter if they try to post anything to reddit, if you can even convince them to get that far.
The platform needs to enforce some type of natural separation. Twitter seems to accomplish this by asking the user what they want to see before they show them anything. That way, people who hate the Pope aren't subjected to his tweets, and people who hate Rachel Maddow aren't subjected to her tweets. Tweets are also discrete units, which makes it so that there is little forced intermingling between hostile parties, whereas reddit articles come with all of the comments and baggage attached, and you can't share them without sharing the comment thread, which invariably across practically every subreddit has a very specific slant that doesn't necessarily correlate with universal values.