"They know I love them anyway"
This is the belief that lets the regret pass by morning. You shout, but you also cuddle, read to them, turn up to everything and tell them daily that you love them. The shouting is a small part of a warm home. Surely the warmth outweighs it.
The warmth is real and it matters. Jennifer Lansford and colleagues followed nearly twelve hundred children in eight countries and found that mothers' warmth went with falling anxiety and aggression over time. Warmth protects children from a good deal. I'm not going to argue that your love doesn't count.
But look at what the research says about whether warmth cancels harshness. Wang and Kenny, in the study from chapter 7, tested exactly this. They expected that shouting would do less harm in warm homes. It didn't. Parental warmth made no difference to the link between harsh verbal discipline and the children's conduct problems and depressive symptoms. Lansford's study asked the same about physical punishment and found, in some countries, something worse. Children whose mothers were both warm and used physical punishment showed increases in anxiety over time. The researchers' advice to clinicians everywhere was to counsel parents against it even inside otherwise loving relationships.
I'd suggest a reason, and it's my reasoning. Warmth and the shout aren't on the same scale, so one can't be subtracted from the other. Warmth tells a child that they're loved. The shout tells them that the person who loves them can, without much warning, become frightening. Both lessons get learned. The second isn't erased by the first. If anything, it's sharper for the contrast, because the child has no way to predict which parent is coming through the door.
There's also what the shout does to the warmth itself. Tuesday's bedtime was gentle, to make up for it. Your child may have felt that, and may also have felt that the gentleness was an apology, and that the two go together. Over years, that pairs love with the aftermath of fear, which is not a pairing you'd choose for them.
So your children do know you love them. That's the good news, and it's the reason the rest is worth doing. It isn't a reason to leave the shouting alone. The love and the shout are both being learned, separately, and you can only remove one of them.