If you have a routing number, you can almost always do a Google search and figure out what bank the money went to. Routing numbers are public information. You won't be able to determine who owns the particular account the money went into however. Your bank won't know who the destination account belongs to either.
If the police get a warrant, they can ask the destination bank who owns the account your money was sent to. Of course, that person is probably not the scammer. It's probably a patsy that was scammed into being a money mule or otherwise unwitting participant in the scam. There is a decent chance that the trail ends with some anonymous transaction-- the money mule bought gift cards or bitcoin or something like that.
As a practical matter, chasing that trail often requires a lot of police resources particularly for the relatively small amounts of money involved. If you're dealing with police in, say, California but the destination bank is in New York, California police would need to reach out to New York police to speak with the recipient. That recipient will almost certainly point the finger at someone else in yet another state. Eventually, you'll find that someone in, say, Kansas was told to convert the money to bitcoin and send it to an anonymous wallet that is likely owned by someone in a foreign country.
As for why your bank is reluctant to give you information, banking information is supposed to be private. The customer service rep may well believe you that you were the victim of a scam but they have no way to verify that. If they could determine who owned the destination account (say because the routing number actually went back to the same bank) and they gave that information out without a warrant, they could be liable if you turned out to be a stalker or if you decided to take matters into your own hands and go confront the person that you sent money to. If the police want that information, they're free to get a warrant and ask for it.