One of the roles of government is to protect the health of its citizens.
Unfortunately the U.S. government has eased into this role like a greasy douchebag sliding up to a dame at the bar, but that is contrary to the foundational principles of the U.S. of A.
The best way to make your case that I can think of is seat belt laws. If you drive without a seat belt you're putting no one else at risk except yourself yet the government mandates it. IMO, the government has the right to mandate that car makers provide seat belts and that they be held to certain safety standards, but I don't think you should be able to get a ticket for not wearing one. Will this lead to an increase in car injuries and deaths? Yes. Temporarily. But people would grow wiser over time and look how successful PR campaigns have been at eliminating smoking! You could continue to educate people on the benefits of seat belts and remind and encourage them to wear them without making it a misdemeanor not to.
When you take away responsibility for personal risk from people this results in less responsible people.
But seatbelts aside, for the most part government's role is not or should not be "to protect the health of its citizens", it is to provide retribution to those who harm others, put others at imminent risk (as in reckless driving, not hypothetical risk through a chain of hypothetical infections), and to create an environment of trust such that I can be assured what you're selling me is legit as described otherwise retribution cometh.
There is a non-controversial level (excluding the most hardened, fundamentalist libertarians) at which it is reasonable and preferable for governments to intervene. And that is the level we are operating at here, when you look at whether or not there should be vaccine mandates.
Since the vaccine doesn't prevent spread we don't have to address the argument of "putting others at imminent risk". We only need to address the seatbelt argument, and here is how the vaccine mandate is different from a seatbelt mandate:
The vaccine is injected into your body and causes physiological changes which will not be the same for everyone, and in fact this puts some people at risk of adverse reactions which can be extremely serious and deadly. Especially for youth predisposed to heart conditions the risks may significantly outweigh the benefits. The risks are known to be there and the full extent of these risks is not yet fully known. Not everyone's body is the same or will respond the same to the vaccine.
The personal freedom of bodily integrity and autonomy to choose to refuse something that gets injected into your own bloodstream should be held sacred.
If you choose to not wear a seatbelt you can still work. If you get caught you merely have to pay a fine. What is being required with the mandates is that you cannot be employed or visit any public place unless you can prove you're vaccinated and in order to prove you're vaccinated you have to have a registry and ID tied to your body. Its as if they used Revelation 13 as an instruction manual here.
Finally, I mentioned one role of government is to create an environment of trust where I can believe that what you're selling is as advertised... well the government has severely failed in this role and destroyed trust in vaccines with the 1986 vaccine injury act which essentially eliminates liability for the vaccine makers and provides them with a free defense counsel and siphons off all injury claims to a separate court. ...and not to mention the revolving door between these industries and regulatory bodies resulting in "regulatory capture".
Leaving vaccination up to personal choice means potentially hundreds of thousands of unnecessary deaths.
You're free to go eat a cheeseburger and fries and twinkies and go skydiving... heart disease and cancer still top COVID deaths. Why don't they just mandate everyone get in shape? This would not only reduce COVID deaths but also the bigger killers. Would be far more effective than the vaccine too. Wait... now that I said that, shit.. that's where this is going isn't it? Joe Rogan doesn't realize he's preparing the way for fitness mandates... they'll soon start requiring everyone to get in shape and we'll all be like Winston with the telescreen yapping at us to do 10 more jumping jacks!
anyway...
This would be like eliminating law-enforcement and leaving personal safety and security up to personal choice.
No it would be like eliminating the seat belt law except that most people would still wear seat belts and most people would still get the vaccine.
Again, law enforcement is mainly for retribution against those who harm others. You can't rely solely on police to protect you which is why here in Texas we like our guns and weapons. Individualist mindset runs strong here.
I'm not claiming that's a bad thing. I just want to make it clear that you are asking government to fail in their duty in this particular case.
It is a pet owners duty to make sure their pets have shots before you take them to a dog park, and that is precisely why people are against this: they don't want their relationship with their government to be that of domesticated animal and master.