Late fees are critical. In some businesses the accounting department will manage debt by paying off vendors in the order of who charges the most interest. If you don't have a late fee, you get shuffled to the bottom of the pile and stay there. Accountants are generally hardwired to avoid penalties, so if they see there are penalties they'll pay before the penalty period (provided they have the money).
However I have found that if your late fee is high, they will simply pay the invoice and "forget" the late fee. And then you're left squabbling over the late fee, which is quite annoying.
Also note that I disagree with OP's interest scheme - in my case its a flat 1-2%. For a freelancer you don't want to tip the scales into lawyer-worthy disputes...