...so the ping times from one side to the other would be 200,000 years
Here, you're assuming that messages and communication has to be two-way. There are a number of really good reasons why one-way broadcasts would be a desirable means of communication.
Here are a couple of scenarios I can think of:
1) Regular status-updates on your local corner of the galaxy. This isn't a strange idea, think of those multi-page holiday update letters you get from distant relatives each year.
2) Announcing the arrival and successful colonization of a new stellar system/neighborhood. The analogy here is the announcement of the arrival of a new baby.
3) Warnings. If you are invaded by a hostile group, you might send out something akin to a warning to allies to cease communications or develop their own defenses. Your message, at least, is likely to get their before the hostile group.
4) Scientific bulletins. Though there's bound to be some duplication, significant discoveries and advancements would definitely be worth sending over the neutrino-wire.
Of course there are many more, and many that none of us could envision because we don't have the experience of maintaining a galactic civilization.