I understand this POV but Cagney's character was so aggressive I'm just a little skeptical that he could contain his urges. Note that High Sierra with Bogart has a similar situation. To me how all of "this" plays out in High Sierra is done in a lot more realistic, as well as, moving way.
I barely remember High Sierra. I really need to see it again.
As for Cagney in The Roaring Twenties, he had no choice but to contain his urges. What if the woman he loved suddenly showed up at his doorstep because she desperately needed his help, and he was at home with another woman? He'd have NO chance with the woman he loved if this happened. If he went to another woman's home for sex, he would have risked missing out if the woman he loved came to his home with some emergency or whatever. I think that, day after day, he waited for her. Later, he was probably kicking himself because, all the time that he was waiting, she was with the lawyer.
On the other hand, like you say, maybe he couldn't contain his urges and the "good girl" knew this, which is why she would never have gotten serious with him.
Personally, I think that in The Roaring Twenties, Cagney did force himself to devote himself to the woman he loved. In The Public Enemy, he wasn't serious about any lady.