Personally, Spider-Man 3 would've been better without Harry becoming New Goblin. If we didn't have the whole "Goblin interference" thing going on, there could have been more to Venom and Sandman's characters. In fact, using the symbiote changes made to Peter without using New Goblin could have actually been good incentive for a reason to become New Goblin.
Basically, think of it this way: Rather than Harry just being crazy-revenge guy, he could be portrayed moreso as a guy who "sees behind the silver lining". Basically, it's the thought process of "I know you're evil; even if nobody else wants to acknowledge it."
If Peter did evil things to Harry when Harry was just his normal self, Norman's goading would be more effective in turning Harry into the Goblin. Harry most likely read about what Spider-Man (in the black suit) did to everyone and may have confronted him about it only to have Peter give him trouble and humiliate him. Given that Harry, himself, was already upset with Peter for "killing his father", the added sight of Peter being a troublemaker and harm to the public would spike more than just revenge and a thirst for blind justice for his own wants.
As for Venom and Sandman teaming up, I think another person or people could step in and show they forgive Spider-Man for what he did. For instance, replace the bombing of Sandman with SWAT or other forces (like Firefighters or Police) trying their hand to distract him from smashing Spidey. Seriously, the public forces have pretty minor roles vs. Spider-Man. I think it's time the people be given a chance to help Spidey as he helps them. The inclusion of Marko's family trying to stop him would work out as it did in the novel, leaving Spider-Man to deal with Venom one-on-one.
I'm not saying to remove Harry completely; I'm just saying that New Goblin didn't really help the film as much as Harry by himself could have.