The standard conception of property in American legal thought is the 'bundle of rights' metaphor: https://en.wikipedia.org/wiki/Bundle_of_rights
The first is that there are a ton of rules about owning property, and (for the most part) nobody thinks that any of those other rules mean that it's not really the owner's property. There are zoning laws, for instance: if I own a house, I can't run a business there. If I own an office building, I can't live there. If I own a three-story building, I probably can't knock it down and replace it with a forty-story building. There's the rule of law: if I run a suspected meth lab in my basement, or harbor a suspected criminal in a spare bedroom, the state gets to barge in. There's taxes: if I don't give the state money merely for the privilege of owning property, it gets to take the property back. There are rent control/stabilization laws: depending on what my property is, I can't charge more than a certain amount to rent it out, even if the tenant is willing to pay. There are certainly political theories under which all of those mean that I don't really own the property, but those aren't the theories we live under. Airbnb shouldn't get an excuse just because we want to start theorizing at that particular point.
The second is that, for most rental property in New York, the state is already involved. There are tax exemption programs, subsidies, city services and infrastructure, and all sorts of things without which the building would not have existed in the first place. The state is more than free to set conditions on buildings where it helped with the construction, even if that condition is "We get to set rules arbitrarily forevermore" - if an owner doesn't want to agree, they're welcome to try building without the state's help.
Think you do? Do you have an alodial title for your property?
No? You are a serf who rents the privilege of a roof over your head from the state. The illusion is nice but it's not real.
Did you say yes to that particular leading question?
Then why say no to renting it out to that same father for 10 days?
Exactly. The properties are being used for exactly what they were meant to be used for, living space.
Your statement amounts to
"can I run around in my backyard firing my weapons off in random directions? Well then why would you expect them to allow me to live there quietly?"
The father probably wants his children to be able to sleep without interruption at the weekend, but is more likely to clatter up the stairs drunk at 3am during his 10-day business trip.
1. "If the state has a say in renting a property then it's not really the proprietor's property".
2. "If the state has a say in renting a property in a way which has no effect on neighbours then it's not really the proprietor's property"
The person you replied to was arguing against proposition 1. You seem to be arguing in favour of proposition 2. Perhaps you are both correct?
Can a proprietor rent out his property to someone else, and possibly so against the will of the state?
I do think yes, if the state has a say then it's not really the proprietor's property.