Government services are not something that are "used" in the normal sense of the word. Central Park needs to get mowed, street lights need to be lit, police and fire protection needs to be provided, etc. regardless of whether residents are there or not.
Another way to look at the issue, is to evaluate the 'tax collection opportunity costs' of the absentee owners. For every condo that gets bought by an absentee owner (as opposed to a local owner), NYC loses income tax, sales tax, and general economic activity (these are wealthy individuals, there spending is a boost to the local economy). By increasing the property taxes, NYC would shift the cost-benefit ratio towards those that are actually going to utilize the property (thus resulting in more income and sales taxes), while also bringing in more property taxes
Finally, if these wealthy individuals were so concerned about only paying for what they "use", then it would make a lot more sense if they just got a nice hotel room for the days/weeks per year that they are in NYC, rather than buying a condo that sits empty most of the time.
And your hotel idea is exactly what will happen if property tax rates go up. Think through what the effects of that would be...
I normally disapprove of using tax policy to modify behavior, but in this case it makes sense. Property tax is like the "rent" citizens pay to the government for using land. If the rent is too low, people overconsume the scarce resource.
Property taxes are relatively low but they do exist. So that billionaire is probably still paying for a cop or a Sanitation worker, even if he steps in NY once a year. NYC takes almost 4% in tax, in addition to what the NY (state) takes.
But that isn't true. Real estate isn't that safe in the first place, and they are also throwing money away by not renting it. They'd do better in buying a commercial property and renting it out.
Vancouver is suffering from this with wealthy mainland chinese, leading to the second worst affordability ratio (housing price : average annual income) in the world. You see similar behaviors in condos in china itself.
It has very bad effects for the people who already live there, especially for the young and newly established. Younger people with better opportunities go away once they do the math and the cities eventually suffer brain drain. I would suggest a high long term unoccupied residence tax to avoid the problems of unused property in cities. Hopefully it will prod these people just a little bit to hire property management companies to avoid the tax and rent the places out.
And really, the better way to look at it is that the property tax paid by some multi-billionaire who only drops into town for a few weekends a year is all profit for the city. They're not using services. They're not sending their kids to public school or generating garbage or calling the police ... whatever they pay to the city each year is just gravy. They're subsidizing people who actually do live there, and would otherwise have to pay more taxes (of one form or another) for the actual services that NYC provides.