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.
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.