Alternative to voltage regulator S81250HG
Given that the 7805 is a bit more powerful than the original device AND there doesn't appear to be problems with low-drop-out in this application, the 7805 should be a good temporary substitute to prove the point: -

At least to see if the device turns on. And then I would look for a
proper replacement.

It may prove to be a bit noisier than the original so that is something that might put you off making it a permanent replacement. Also, the 7805 does appear to be more hungry in terms of quiescent current so, if the unit is battery powered this may also cause a problem. I couldn't check that because the link to the product is faulty.