First, it's not just about the current bailout. It's a pattern of behavior going back to the eighties, and especially evident after the re-union with East Germany, and it has to do with control of EU directives and policies (including, but not limited to, monetary policies).
(Or rather, one could say, it's a pattern of behavior going back to the first and second world wars, as the whole European Union, as an initiative was thought up to constrain future German foul-play ( http://www.amazon.com/The-New-World-Perry-Anderson/dp/184467... )).
>It's Germany's money that's bankrolling the bailouts, and as far as I can tell
Not really. In fact they could care less about Greece or the South in general, it's about saving their banks and using tax-payer money to siphon it to the German (and French etc) equivalent of Wall Street:
http://www.economist.com/node/18560535
http://speri.dept.shef.ac.uk/2013/11/11/dirty-secret-euro-zo...
http://www.bloomberg.com/bw/articles/2013-09-12/is-germany-r...
http://www.bloomberg.com/news/articles/2013-06-11/crisis-sav...
http://www.spiegel.de/international/europe/germany-profiting...
Brexit is a political decision. It's not inherently good or bad, some people with so and so interests find it good, others with so and so interests find it bad. Similarly, some people think other people voted that "against their own best interests", other people think that they voted according to their interests just fine, and that the condescending ones don't really know better.
Countries signed up to the EU because of certain proposed benefits (among them, the idea that they need to keep Germany bound, an openly stated goal of early diplomats and politicians that were fundamental in the creation of ECSC/EEC/EU. Long after that, Germany re-united, took the de facto economic leadership of EU, and now votes for AfD.
(For a great overview of the climate and the tensions behind EU's creation and later development, I suggest: https://www.amazon.com/New-Old-World-Perry-Anderson/dp/18446... ).
Furthermore, EU has always been problematic from a democratic perspective, with bureaucrats at the helm, and an opaque process of decision-making (complete with several non-official-bodies, non-accountable to anyone, that nevertheless convene and take decisions that affect all member states). Add to that certain "circles of influence" between countries due to historical, cultural and economic ties, where a country might control several other countries votes (and thus have undue influence in EU decisions) and it's not that great a bundle deal.
So, if some country doesn't see the benefits in EU membership, they can always flee. Whether it will hurt its economy is not the be-all end-all question to make the decision "rational" or not, as a) the EU is not exactly a paragon of economic stability, b) economies can rebound, c) the economic impact is not the ultimate criterion by which countries decide everything they do. (As for an open market and open standard they can always happen across European countries without common single law and economic steering from Brussels -- I don't think China and the US for example have much issues in trading, despite not belonging to the same federation).
It's also about personal interests. Young voters might vote Bremain for reasons like "it's the progressive thing to do", "we want a united Europe", "I want to be able to work in any EU country" etc. Older voters might be more cynical on to what EU actually offers because of their experience, and they also don't have much to gain by EU mobility (nobody --and by that I mean extremely few-- is going to leave everything and start a new career in France in their 50s and 60s) or other things that might be enticing to young voters. That might be selfish of them, but it's not senile or irrational: it's pragmatic.
In other words, no, Brexit is not some "proof" that older British get increasingly unwise. At worst, it could mean that they can be wrong in some decision, as is the case with everyone. But the question was not whether they can never cast a wrong vote, but whether they have more experience, control over emotion, better decision making empathy, and other such traits TFA talks about than their younger cohorts. You can have all those and still make this or that bad decision.