Public health officials should tread carefully otherwise they risk alienating large segments of the population. Which would inturn be bad for public health.
Already happened. The resistance to a lot of the public policies in NA (both US/Canada) during the pandemic is directly due to instances of overreach such as this.
In Australia the government has largely lost control of the population. They tried to lock down hard and faced resistance but every time they try to crack down harder or enforce the rules more, they get bigger protests.
Now they have just given up entirely and even the most minor restrictions and requests are largely ignored.
Do you live in the same Australia that I'm living in? Because that's not what I've observed at all, at least in Victoria.
For better or worse, the vase majority of Australians have complied with whatever restrictions and mandates that governments have put in place. VIC and NSW, which have borne the brunt of cases and broadly mandated vaccination, have ~95% of people aged 16+ vaccinated. People have generally complied with lockdown restrictions and requirements to wear masks.
Despite the media attention they garner, the protests involve a small minority of people. Victoria didn't see any significant opposition to lockdowns until the later half of this year. Most people seem happy enough getting paid $700 a week to sit at home and play Xbox.
That said, I have noticed that increasingly people are not bothering with the more tedious requirements such as wearing masks and checking in. I wouldn't say anywhere close to "largely ignored" though. I get the impression (from people I know) that it's mostly due to fatigue and risk habituation, rather than alienation.
Australia _has_ had pretty good compliance but I don't think they have control anymore. I don't think any state has the power to pull another lockdown, it just wouldn't go down with the public. When I was in Melbourne last week, mask rules were frequently ignored, QR codes entirely ignored and the majority of non essential businesses did not check my vaccine proof. I also had business owners tell me how they hate the government forcing them to check the vaccine proofs.
Vic tried to pull tighter restrictions which resulted in mass protests and the PM saying something along the lines of "The people have had enough of restrictions", after that restrictions have been going away.