I don't know about CLE specifically but while the airport is self-sustaining, all kinds of tricks can be used in other cities on the "subsidies" front -- airport revenues (as you pointed out -- these are not unrestricted tax dollars), Convention/Visitors Bureau, Economic Development Corporations, city/county/state grants, even individual corporations etc.

[It is also worth noting that the "must be spent solely for airport purposes" is a federal requirement for any airports that have accepted federal monies for an airport -- which I believe is just about every airport with commercial service in the US. This is to prevent cities from "milking" their airports for local projects. On the other hand there's no requirement that cities cannot put money *into* an airport if they so desire]

Personally without some expressed level of demand I think subsidies are wasted feel-good money. Sure we could give Qantas $100 million or whatever to start/prop up nonstop Cleveland-Sydney service but after a month or two if there aren't enough passengers at a high enough fare either the service goes away (loss of initial subsidy) or "we" (the City, etc.) dump more money into a proposition that is not sustainable.

Carefully used, of course, subsidies can be a powerful tool... but "me want International!111!" is not a strong enough reason on its own.