Here's a wrench I don't think I've seen thrown into this conversation yet: how useful parsers can be for finding bugs and balance issues. Blizzard is constantly patching weird, obscure bugs in spell interactions and making small tweaks to balance because players running parsers find things that testing misses. Any developer worth their job will tell you real-world tests are going to find stuff that your testing environments won't. Users will always do things developers and designers don't think of. And letting users have parsers makes that worlds better. Everyone feels their DPS or threat or HPS or whatever is too low or too high, or swears they're missing when they shouldn't be or that an ability isn't quite doing what it says on the box.
Without parsers, it becomes incredibly difficult to make the case for any of that, because it's just vague forum noise with no real data to go on, and likely to be drowned out by "It's fine, shut up." With a player base armed with parsers, you get enough actual data to see whether it's true or not.
Without parsers, you end up with years of "ROFLDRG" because everyone assumes their DPS is crap, and there's never anything solid to make them see otherwise.