Luke – as always, it depends.
If the pipe was within or on Common Property but it served only your Lot, then its repair and the rectification of any damage caused by its failure are both your responsibility, with the location of the problem section of pipework and the rectification of any damage to the Lot being claimable by you on your contents insurance.
However if the pipe served multiple Lots then its repair and the rectification of any damage that may be caused during that (repair) activity, but not by the burst pipe itself, are both the responsibility of the Owners Corporation (O/C). Similarly your Plan’s building insurance would, if your O/C chooses to claim, cover the costs of locating the problem section of pipework and the rectification of any damage that the burst pipe caused to Common Property.
As for the O/C checking the integrity of its pipework, whilst that can be done the precise identification weak / leakage points is marginal, and is in my opinion beyond what could reasonably be expected and above what’s required of an O/C under Sect 62 of the NSW Strata Schemes Management Act (1996).