Where in the Constitution does it say the state has a right to a fair trial?
You are not asking the correct question.
The State has an obligation to ensure a fair trial, as an unfair trial, is reason to have a guilty verdict thrown out by Appellate courts.
As you are pointing out, the defendant has a right to a fair trial but the defendants lawyer not only has no obligation to make sure the trial is fair, it is a great back up plan for the defense to ensure it is not fair, so if they win their client gets off, and if they lose, their client can have the verdict thrown out on appeal.
What does that mean in real life?
Take the trial of Walt Nata in Judge Cannons court where Jack Smith is demanding a Garcia Hearing, to determine if Walt Nada's lawyer has a conflict of interest that could harm the defense of Nada.
No prosecutor necessarily cares if an accused is getting poor and inadequate legal advice except for the fact, that if the person is found guilty, and appeals on the basis of 'Inadequate legal advice', that appellate Court hearing will ask the prosecutor 'was there a potential deal on the table for this convicted person', and if the prosecutor says 'yes', and then they ask the defense lawyer 'did you ever let your client know there was a potential deal for them to consider' and they say 'no', that can and does get convictions vacated. It is a back door way for a defense lawyer to get his client off, in a case he lost.
So that is why the Prosecutor forces the Garcia hearing, to PROTECT the defendant and the Fair Trial. In the Garcia Hearing the prosecutor will say in front of the defendant everything that is available to him, so he cannot say he does not know, and then if he still wants to stay with his current lawyer he can.
So YES the State is OBLIGATED to ENSURE a fair trial, as an unfair trial is reason to vacate a guilty verdict on appeal and EVERY defense lawyer knows that and it would become a checkmate tactic in ALL trials.