I don't think they should.
Most independent media is just worthless opinion columns, political activism made to look like news, and on some occasions just straight up disinformation.
Sure some of it may be ok, but if you try to write legislation that comes out as "all left leaning independent media gets money, all the right wing independent media can go pound sand" it's just the government trying to use legislation to promote their party. That's a really bad precedent.
So as much as I'd like to see the good independent journalism funded by this, it doesn't seem feasible to do that without also funding disinformation.