Is a 75 360 LA supposed to have a oil slinger?

It should have an oil slinger.
That is not a huge amount of play in the chain, I believe the factory service manual says at 1/2" slop to replace. The main thing is the cam gear steel?
LOL.. if you are any where near to the factory service limit of 1/2" on timing chain slop you have around 4 degrees of cam retard! I have measured it.

I personally would replace the chain and sprockets at this time since it is open unless this was just a beater car.