Welcome to Rootschat from me too, Ev

I hope you don't think poorly of the oral history that has been passed to you. There's usually some sensible explanations that were significant to the people who first heard or passed these on.
Many families had secrets, sometimes because they were avoiding the possibility of being 'found out', either for apparent violation of an unwritten social rule, or for the more serious violation of the criminal code ...
As I understand it (I've got several 'bigamists' in my ancestors) Police officers did not go around specifically looking for proof that a person was or was not a bigamist even in the 19th Century they only acted on that particular violiation of the criminal code IF they were given valid information ... usually this would be at least a Stat Dec signed by one of the spouses (who needed funds/maintenance to continue to support the children of the marriage) ....
However, from around the late 1870's with changes to Divorce Laws (NSW c1875) there was the fear of being thought of being bigamous. This 'fear' was often the reason for couples moving to a different locality and thus "co-habitating" as though a married couple. The 'fear" was thus avoided : - often the couple would 'co-habitate' as opposed to having the earlier marriage dissolved, and thus they avoided being mentioned for listing in the Law Notices in the newspapers. If either party brought children of an earlier marriage to that new relationship, then the children's surnames became that of the new male partner... So they enrolled at their new school with their new surname, they grew up, with that new surname, married with that new surname ... It was (and still IS) LEGAL to change your name ... either simply by usage or by formally applying to change it.
Women of my generation who are/were married usually changed their surname simply by using their married surname. I know that I produced my marriage certificate to change my drivers licence, but the bank didn't need any documentation, a family member who attended my marriage also worked in the bank where our accounts were then held. In those days the "KTS" initals would have been good enough ! (KTS = Known To Staff)
You asked "why else would they go into hiding" .... Perhaps they were not 'hiding' but just getting on with their own lives ... The tyranny of distance, lack of communications, poor reading or writing skills by earlier generations, lack of good knowledge of the full (postal) address or current surname of a family member ...
Sorry for the long reply,
Cheers, JM