Dr. Ford did request an investigation several times. She also understands that lying to the FBI is crime. When Cory Booker asked Kavanaugh if he too would welcome an investigation, Kavanaugh deflected the question several times and never gave a straight answer.
Dr. Ford came forward amid all the backlash she and her family are going to receive from rightwingers. She literally has nothing to gain from this. Kavanaugh is going to be just fine. He has tons of connections, and lots of money. Not getting a seat on the Supreme Court is hardly ruining his life or even his career. At the most it’s a bruise to his ego. That doesn’t elicit pity from me, no matter how much he tries to come off as a victim.
The smartest thing Kavanaugh could do is acknowledge any wrongdoing and apologize for any harm he has caused. Unfortunately he is too arrogant and self-centered to do that.
You are correct, it is the President’s constitutional duty to nominate Supreme Court justices, and it’s the Senate’s duty to fully vet that nominee, or it was until Senate Republicans refused to even consider Merrick Garland after Obama nominated him, and the court had to function with only eight justices for a year.