The situation at the ATMs is not because Greece is spending too much, on a primary basis.
It is because the ECB removed ELA support - or rather kept the same cap. This is more about the liquidity of the banks in the face of a bank run, not capitalisation. The Greek banks were recapitalised a few years ago, so they're meant to be ok.
The bank run is because people think there is a real possibility of Greece returning to its own currency (which probably would overall be better for Greece, given the politics in the EU). It makes sense to get your money out of the bank in a hard currency like the €, rather than wait and risk your deposits being converted to drachma by a new law. The bank run is *not* because Greeks are afraid their banks are about to run out of money - they're well capitalised now AIUI (least, as well capitalised as most other EU banks).
Note, if for some weird there was a bank run on German banks tomorrow, they too would have to close, outside of special liquidity support being given.
This is a basic function of traditional banking: a bank lends out or otherwise invests the money deposited with it. So a bank generally doesn't have enough cash (itself) to pay
Which is why we have central banks, to support healthy banks (i.e. well capitalised) through any liquidity needs. Which the ECB has stopped doing for the Bank of Greece, and hence the general Greek banking sector.