I think this is a good idea, but caution should be taken to not punish players who have reasonable reasons to rage quit a lobby.
For example, I had to rage quit two lobbies because the server being used was giving me a ping of 850ms and the game was unplayable. I did this before round start.
It needs to be acknowledged that there isn't currently a way for a player to back out of a lobby in a controlled manner. Reporting yourself or having others report yourself obviously has (and should have) a stigma to it worse than a rage quit.
The above problem could be fixed by offering a command to forfeit for your place. Forfeiture should still not be something which can occur on a regular basis, but there are ways to control for it. For example, you are awarded one forfit point per ten lobbies played, with a max of five or ten forfeits, and then they count as rage quits.
The auto reporting system takes awhile to kick in, and it takes time away from the whole team/server to have to rep a person when they leave mid-match. Being able to forfeit your spot saves everyone time and would get a replacement into the match faster.
Is this a problem that the same players are serially rage-quitting, or are there enough players in the general population rage-quitting? Depending on the answer, there are different ways to address the issue.
Anyway, back to the issue of addressing rage quitters themselves...
Serial rage-quitters rage quit a lot. It should be easy to identify them.
Here are two examples of two rage-quitters which I have seen playing in the last two weeks. These players have rage quit from at least three lobbies which I have been in personally, and I've seen them being called out for their behavior in lobby chat. They basically have a bad community reputation now.
https://tf2center.co...561198165163405
https://tf2center.co...561198025469129
In the case of Whopper, he was up to a 25% rage quit rate with 10/40, within a period of 30 days. Easy to spot.
In the case of Loveclop, he has far more total lobbies played, but I am willing to bet his recent percentage is very similar to Whopper there.
Here is another idea: If a player reaches a certain rage quit percentage, make them click-through a warning page to be able to browse lobbies again.
This won't work for trolls, but I am willing to bet it could have a positive impact for most instances.
If players know their rage quit count is being watched and are warned over it, that could modify behavior.