I do not think this is true at all. Fear is not an effective tool for improved behaviour according to behavioural research. Punishment on its own is shown not to improve behaviour, you have to have other aspects such as clarity, consistency, and some positive aspects like good modelling from the parents or encouragement of the positive behaviour (rewards, praise, attention, whatever). It can have an effect when paired with at least some of these aspects, but the severity of the punishment makes no difference, ie being shouted at/hit is no more effective than e.g. receiving a small sanction such as losing a point on a chart.
What you will often get if you invoke credible threat (e.g., threatening a child with physical punishment or something else they are genuinely scared of and believe you will do), and what is often mistaken for the idea that fear or harsh punishment "works", is compliance with demands and expectations in the moment, but not generally better behaviour overall, and it's not even especially clear that this shouting OP describes is being used in that way, which might mean that it is not even effective as a punishment because it's not clearly linked to the behaviour (and I don't mean that all punishments need to be related as is the current fashion - but for example if you place a child in time out and tell them this is for not putting on shoes when told, or even - I wouldn't do it, and the research is out of date, but if you smack a child and tell them that is for running in the road.) Just shouting and ranting, even if the content of the shouting contains the reason, is unlikely to register very much because it will push a child into more of a fight/flight/fawn response and they don't have as much access to the language/learning centres of the brain at that point, so they are probably registering more their own fear and upset rather than whatever it was they were doing before which triggered the outburst. So it is unlikely to pre-emptively change their behaviour in the future, which might be why OP remembers the "not hugging goodbye" issue seeming to be a repeated scene.
In addition there is some evidence that parents who use very impulsive, emotion based strategies to discipline such as angrily shouting may have poor impulse control and emotional regulation, tendencies which can be inherited, so children with these tendencies might also be less able to self-regulate their own behaviour particularly when their main adult role model is failing to do so.
Another thing which can happen is that if the children begin to associate the angry parental shouting with a certain task or part of their routine, it can introduce more generalised anxiety around the entire routine which can worsen in turn the children's behaviour (because anxious children have less impulse control/emotional regulation). So it could be that the children even start to play up more around the time of leaving because they anticipate that leaving tends to lead to a point where their dad is shouting and scary. And dad has these memories of it being stressful and embarrassing so he is also on edge for the children to start playing up and is perhaps quicker to jump in and have a go at them. It's a really destructive self-feeding cycle.
Actually, OP, if you think he might be receptive and you don't want to refer it to any authority just yet, you could possibly send him a copy of the book 123 Magic, saying that you keep hearing friends mention it and absolutely raving about how helpful it was, or something like that, saying something noncommittal like "I don't know if it's any good but it sounded like something which was worth a try if you wanted to". It's a good book when parents are being too harsh, because it's simple to follow and it explains why things like shouting don't really work and tend to lead to more of the same behaviour rather than less, and it gives you a concrete thing to replace it with which generally does tend to work, and it has been used in programs in the US where abusive parents are court ordered to do a course before they are allowed to have contact with their children (something I'm not sure I agree with, but anyway).
It's not as "fluffy" as a lot of books with a more positive or gentle parenting type lean, so it tends to appeal more to parents who would be put off by those things, and TBH it's not my favourite book because I think it leans too heavily on punishment and compliance, personally. But a calm time out, which is what the book advocates for, backed up with loss of privileges if the child doesn't comply with the time out, is much better than being shouted at all the time. Even people who disagree with the concept of time out tend to agree that it's better than shouting.
If they did change to following something like that I expect it would reduce the stress levels in the home hugely so they would probably have more energy and emotional space for more positive/fun connection stuff with the DC. Two under 5 are stressful ages and it's probably much worse if you only have ineffective and escalating strategies to try and manage their behaviour.