From my own limited experience, my input is this: I think there are two main things that should be focused on to help bring in new comp players.
1. Getting them excited about it.
Streams are great for getting people acquainted with each other and high level players within the game, but things like good casts, povs with comms, etc. are really exciting and interesting to people who are new to comp, or even to TF2 in general. Unfortunately they are currently just way too disorganized and hard to find, especially for people who are just trying to get into it and track a team or player's performances. When they are able to do that, they can really get sucked in fast.
I think there needs to be a database of match casts and player POVs that can be tied into a schedule of matches, and team/player profile pages. Newbies won't be on the STV/Mumble right away, so it's important to have easy access to past matches.
So, for example, someone can look at a match schedule and click the info for an interesting match and see the video/video links, then they watch some and say, "wow, that one team is great" and click on their team profile, and get links to all their past matches with videos. Then they go, "damn, that one player keeps making huge plays in all these matches" so they go to his profile, and get links for all his previous POVs with comms, etc. Quickly they can become an excited, invested fan.
2. Making it accessible to them easily and with low pressure.
A certain percentage of the fans made above will become very interested in trying it out for themselves. To get them in for the long term, there needs to be a low barrier to entry, and a clear path of progression. I think the best idea so far is having a tf2lobbyesque solution for teamfortress.tv. I do agree with people who say the skill level among even the current casual low-level competitive community would be too high for the majority of new players, so either it needs to get big really fast, so the average level goes down, or there needs to be some kind of difficulty/level/skill/whatever tracking system, so that steel-level people don't wander into an invite lobby unawares and have their competitive hopes crushed right off the bat.
The same system could also (as far as i know, having no technical knowledge of it) be used for scheduled newbie mixes with mentors on each team, and even newbie mix lobbies, which could start up when 2 admin mentors create a lobby together or something.
I think we have a very passionate community, but our technical resources and tools for growing it are fragmented and not very advanced at the moment. It's ok for people who have spent the time to dig into and get used to it, but it's a high barrier for people who are completely new to competitive tf2, or who may have a casual interest that a less intimidating set of resources might nurture.
In any case, thanks to tf.tv and the people in here for making strides in the right direction.
Quick mockup for a visual idea of some of the things I'm talking about: