It's not that fragile, although it should of course be improved further. Payment terminals don't depend on constant connectivity, they can store transactions and send them to the bank later. Some of them, especially the ones used by smaller merchants, are battery-powered. If card terminals are unavailable completely, mobile payments should still work. So in my experience, the cashless society here can deal with such intermittent technical problems just fine.

A lunch restaurant near my office had a total outage once, couldn't take any payments. Things worked out just fine on an honor system then - they wrote down names and amounts on paper, and asked people to pay in the next days.

What doesn't work currently is resilience against long-lasting technical problems. If there's no power in an area for several days, there's no sensible way for the current cashless technologies to work.

Cash is not very useful if the power is out either, as for internet you just have to make it as reliable as power.