To quote a different conversation, it's not just size but what you do with it.
Banquet rooms pay for themselves, dining rooms (especially lots of small ones) do not. If they are modest, pools, tennis, and fitness rooms pay for themselves with an addiitional members (and dues). Where clubhouses get killed is with lots of completely unused space that costs money (to heat, cool, and repair), over the top pools, tennis and fitness that cost more to maintain than they generate in revenue, and rediculous finish costs that result in more upfront costs and higher property taxes.
I was recently at a club that had a 50,000 sf clubhouse. It had 8 dining areas, but the largest one could seat no more than 70 people. They have to hold weddings in a tent on the lawn, and were considering expanding the clubhouse. If the club was 100 or even 40 years old, I could understand it, but this one had just opened. Now they have to heat and cool all of these spaces (that are not being used), plus there are two kitchens that need to be staffed, etc.
That said, I'm not going to jump on the bandwagon and blame all of the problems on the clubhouse. There are just as many instances of money going to waste on the golf course. The article even references an average golf maintenance budget of $1.5M, which is absurd. Plus, there's a lot of costs beyond just the golf course maintenance that you have to pay for whether you have a country club or a golf club. Pro shop staff, property taxes, utilities, insurance, repairs, etc. In general, I would say clubhouses and amenities don't have to be a drag. Its usually politics, poor management, or in the case above, poor planning, that often make it so.
All that being said, my favorite clubhouse is the one at Newport Country Club which is essentially a bar and a locker room. And a beat up one at that.