I've never asked an authority) that when something happens that might have the potential for trouble, the first thing you should do is file a police report, just so something is on the record in a timely manner.
The mysterious "they" say a lot of things. Calling the police is one thing. This does not guarantee that a report can or will be generated. Even if one IS generated, there is also no legal requirement that a complete and thorough investigation be done.
I see a potential misdemeanor for the unsafe discharge of a firearm, and maybe property damage (the dog) or animal cruelty ... assuming the shooter cannot properly articulate a fear for their safety or the safety of others.
The law rarely requires an officer to make a report or investigate.
But now it sounds as if police departments can refuse to take or file reports, maybe even in a situation like this? I don't understand.
It's a matter of resources. The police cannot afford to write out and investigate incidents for which there is either no crime, or no real likelihood of prosecution. There are many agencies in my state that will not take property crime reports because of staffing cuts. Others, such as mine, will take the report, but the DA is so short-staffed that they won't file on them.
So, agencies take those reports they can deal with and those that it is worth the expenditure of resources to pursue.
That being said, I would have written something up on the matter - and it's possible that the officer did. It might be he just didn't get your husband's statement.