She can't not tell people if there are children involved, they may need a bit of extra support, f or a while,even if the split is amicable. It'd be cruel not to let their school/nursery know as they are bound to be feeling confused at the very least & it may impact their behavior & concentration. Any caring school would want to know what's happening in order to help them.
Then we move onto finances - she can't be left in limbo so the utilities/bank/child tax credits etc, etc need to know. Buggered if I'd want to retain any shared accounts with a man who had "secretly" left me. The last thing she needs is any stupid beuracratic jobsworthy type issues right now from the taxman or DWP etc.
Likewise his parents have the right to he no longer resides with their grandchildren, along with any fave aunts/uncles from his side.
Friends are probably the last on the priority list of "need to know" people, but again, if he's never in when they call from now on, it's only to coutesy to have the decency to tell em, there's no point popping round to see him at the marital home. She's now his ex, not his mother & those kids deserve honesty. If his friends think less of him, well that's really not her concern.
As an adult he needs to accept responsibility for his own actions, and she hers. It's not her job to shield his licckle wicckle ego, however it is her job to continue being THE best mother to her kids she possibly can, and that includes seeing a solicitor to ensure that they retain the roof over their heads right now. She didn't create this mess, & can only pick up the pieces of it that she retains personal responsibility over.
Sadly my suspicion is that he's trying to shield the OW, who comes from within that mutual circle of friends right now as opposed to prioritising his own bloody kids emotional well being. With a bit of luck the OW's hubby will wake up, smell the coffee and lamp him one 