Does using MyMonero online wallet and sending it to yourself (churning) help increase privacy?
Leaving aside MyMonero online wallets, the benefits or churning are debatable. There are really only a couple of very specific cases where churning can be beneficial (such as breaking the linkage between specific tx outputs, i.e.
seep_single, [ref]), but naively just churning all the time, in most cases actually just reduces privacy [ref]. The best advice is to just use Monero normally, as intended.
I hear people say churning is good for the network.
This is nonsense that was peddled by one specific group of people. The benefit to the network is soo small it's utterly insignificant.
But online wallets do sacrifice some privacy for convenience.
This has nothing to do with churning. Online wallets have numerous downsides, none specific to "churning".
MyMonero has users' public addresses and view keys. Does churning do anything then?
Again, not specific to churning, but you are taking huge heavy privacy and security risks using online wallets disclosing any information to them. The moment you are trusting some service with your wallet view key, wallet address and your IP address, you've already obliterated a huge amount of your privacy. Thus whether you churn or not at this stage is largely irrelevant.