But wouldn't they still investigate to want to prove that Batman did it though? If they find him, and that's all they have, then they have no proof, cause they were only interested in finding him rather than building a case. Wouldn't they want to get actual proof, to make a case, if they did find him?
As for Gordon acting as a witness for Batman murdering Dent, that didn't need to happen either. All Gordon and Batman had to do was get rid of Dent's body so it wouldn't be found, and no one knows what happened to Dent. Maybe he got blown up in the hospital. Maybe Joker killed him in the hospital and left him to blow up. Dent can still remain a hero, and Gordon does not have to explain what Dent's dead body was doing there if him and Batman got rid of it.
Or they could just leave the body there, and lead the public to believe that someone may have killed Dent in all the chaos, but I didn't think it HAD to be Batman.
Right then...
Gotham needed something to unite the city.
The death of Bruce's parents spurned the city and the wealthy, to act upon crime in Gotham. In the end, it didn't work.
So, upon Dent's arrival in Gotham, Bruce could see he was the right person for the job of ending crime.
Dent made a huge impact on the crime syndicates within the city, he had an impact on the People too. Gotham loved Harvey Dent... but Joker played games with everyone, especially Dent.
Joker twisted Dent into a madman. He injured him, and killed his fiance.
Joker then let his new creation (Two Face) loose on the streets.
Dent killed cops, gangsters, and was going to kill the Commissioner's family, and the Commissioner, and Batman too.
Dent had been broken.
Gotham's "White Knight" was broken.
In the struggle with Dent and Batman, and Commissioner Gordon's son, Dent is killed.
So, instead of allowing the truth come out that Dent was probably clinically mad, Batman stepped in to be the punching bag for Gotham.
Blaming Batman for Dent's crimes, and Commissioner Gordon acting as a witness to Batman murdering Dent, means Gotham has something to unite for.
Their White Knight was dead, they needed something or someone to blame, to unite the People.
Batman is that thing, that someone.
Dent is the hero Gotham needed... Batman is the one Gotham deserves but doesn't know it.
In Dark Knight Rises, all this is explained by Gordon to John Blake.
"There's a point. Far out there. When the structures fail you. When the rules aren't weapons anymore, they're shackles, letting the bad guy get ahead. Maybe one day you'll have such a moment of crisis. And in that moment, I hope you have a friend like I did. To plunge their hands into the filth so you can keep yours clean."
Edit/addition: They also couldn't blame Joker for Dent's death, because Joker was already captured by the time of the Dent/Batman/Gordon fiasco in that construction site.