The 5e PHB also told players to check with the DM for changes and the rules gave the DM veto power, but there were still tons of players who felt that if it was in the PHB, they could just pick it. If the DM said no, he was big bad wrong funning them.
Putting it in the DMG makes it crystal clear that the backgrounds, which have a greater potential to fail to meet the DM's setting than feats, races, etc., are DM granted, rather than DM vetoed. It's better if the DM has to say yes for something to happen, rather than saying no to keep it from happening. There are fewer hard feelings that way.