Here's my read on Jack Drake, as a freestyle personal interpretation:
He probably thinks he's a good father because he provides Tim with a comfortable home and a good education, but doesn't put much more effort in than that due to being raised to believe that a husband and father's role is that of a distant breadwinner who gets to enjoy the perks of having a kid to play/bond with on the weekends. While Janet was alive he leaned on her for the majority of the parenting, and after she died he struggles with Tim's independence and disobedience, falling back on boarding school to provide the discipline he can't. That later point is further exacerbated by...
He also probably thinks he's a good father because he doesn't beat his child, but doesn't consider childhood spanking to be the same thing. In his mind, spanking a child is "Necessary Discipline," which makes it okay, while hitting a teenager in anger would be abuse. Screaming at and belittling them before destroying their belongings, though, that's appropriate "discipline" for his teenager.
He probably thinks he's a feminist because he's for universal suffrage and bodily autonomy, would never lay hands on a woman, and was attracted to and married an outspoken go-getter career girl. But he was also disappointed when that career girl, Janet, didn't want to "settle down" and become a good stay-at-home mom after she bore their child. Her outspokenness meant she'd push back against him when he expected to be obeyed as the Master of the House, leading to arguments and, if she'd survived long enough, a likely divorce. And if she had wanted to abort that child so she could focus on her career? He would've blown his top, because it might be her body but that's his child.
He probably thinks he's tolerant because his company had diverse hiring policies and he doesn't personally believe gay people should be stoned to death. But he's not actually the one who instituted those hiring practices, he never would've thought to do so, and also he uncritically believes that the "bad neighborhoods" of Gotham are all the places where people of color live, and that gay men holding hands should be considered public indecency.
As a result of the above, he would probably struggle to accept things when he learned Tim was queer, and while he would probably come around to begrudgingly accepting it in the end, it would only happen after he worked his way through a violently negative initial reaction. I've always thought he'd react exactly the way he did when he found out about Robin -- he'd find someone to blame for "corrupting" his son (likely Tim's boyfriend) and go after that person with a gun with the intent of scaring them off, and then he'd essentially demand Tim go back in the closet and impose a bunch of new restrictions to make sure he stayed there.
(If you want to get extra spicy, Tim only dated Black women after Jack's death. Do you know how many white parents think they're anti-racist until their child brings home a Black partner?)
He would, eventually, be forced to accept the reality of his situation, because he does love Tim and doesn't want to lose him, but it would almost certainly be another thorn in their relationship.
In short, he thinks he's a good person because he gives lots of money to charity. And he does do that. He also self-funds archeological expeditions of dubious ethical origins and thinks the activists lobbying him to stop are a bunch of unemployed hippie losers with nothing better to do than harass him.
He's a typical white conservative American everyman, in other words. Not the world's worst father -- in the DCU, he probably doesn't even make the top 100 worst dads, given all the unethical science-daddies and actually abusive villain origin stories out there. But he's also not a good dad. He's a normal kind of bad parent. Nuanced, human, and normal, but still not great or even good. He's mediocre at his best, and he's usually not at his best.
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